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They point out that real numbers can't capture the distinction between certain extremely unlikely events and genuinely impossible ones—they are both represented by credence 0, which violates a principle known as “regularity.” Following Skyrms 1980 and Lewis 1980, they recommend that we should instead use a much richer set of numbers, called the “hyperreals.” This essay argues that this popular<span id="EASRAI-absexp"> (<span class="ll" onclick="$("EASRAI-abstract2").show();$("EASRAI-absexp").hide()">...</span>)</span><span id="EASRAI-abstract2" style="display:none"> view is the result of two mistakes. The first mistake, which this essay calls the “numerical fallacy,” is to assume that a distinction that isn't represented by different numbers isn't represented at all in a mathematical representation. In this case, the essay claims that although the real numbers do not make all relevant distinctions, the full mathematical structure of a probability function does. The second mistake is that the hyperreals make too many distinctions. They have a much more complex structure than credences in ordinary propositions can have, so they make distinctions that don't exist among credences. While they might be useful for generating certain mathematical models, they will not appear in a faithful mathematical representation of credences of ordinary propositions. (<span class="ll" onclick="$("EASRAI-abstract2").hide();$("EASRAI-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-EASRAI"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/betting-interpretations-and-dutch-books" rel="section">Betting Interpretations and Dutch Books</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/conditional-probability" rel="section">Conditional Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/degrees-of-belief" rel="section">Degrees of Belief</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','EASRAI')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=EASRAI&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1215%2F00318108-2366479"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/EASRAI">(3 more)</a>   <div id="la-EASRAI" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('EASRAI')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-EASRAI" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('EASRAI','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/EASRAI"><i class="fa fa-share-alt"></i> 70 citations</a>   <span class="eMsg" id="msg-EASRAI"></span></div></div></li> <li id="eMANWIT-6" onclick="ee('click','MANWIT-6')" onmouseover="ee('over','MANWIT-6')" onmouseout="ee('out','MANWIT-6')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/MANWIT-6"><span class="articleTitle recTitle">What is the upper limit of value?</span></a><a class="discreet" title="View other works by David Manheim" href="/web/20230531191753/https://philpapers.org/s/David%20Manheim"><span class="name">David Manheim</span></a> & <a class="discreet" title="View other works by Anders Sandberg" href="/web/20230531191753/https://philpapers.org/s/Anders%20Sandberg"><span class="name">Anders Sandberg</span></a> - <span class="pubYear">manuscript</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">How much value can our decisions create? We argue that unless our current understanding of physics is wrong in fairly fundamental ways, there exists an upper limit of value relevant to our decisions. First, due to the speed of light and the definition and conception of economic growth, the limit to economic growth is a restrictive one. Additionally, a related far larger but still finite limit exists for value in a much broader sense due to the physics of information and<span id="MANWIT-6-absexp"> (<span class="ll" onclick="$("MANWIT-6-abstract2").show();$("MANWIT-6-absexp").hide()">...</span>)</span><span id="MANWIT-6-abstract2" style="display:none"> the ability of physical beings to place value on outcomes. We discuss how this argument can handle lexicographic preferences, probabilities, and the implications for infinite ethics and ethical uncertainty. (<span class="ll" onclick="$("MANWIT-6-abstract2").hide();$("MANWIT-6-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-MANWIT-6"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinite-value-theory" rel="section">Infinite Value Theory</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/value-theory-miscellaneous" rel="section">Value Theory, Miscellaneous</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','MANWIT-6')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=MANWIT-6&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FMANWIT-6.pdf"><i class="fa fa-download"></i> Direct download</a>   <div id="la-MANWIT-6" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('MANWIT-6')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-MANWIT-6" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('MANWIT-6','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <span class="eMsg" id="msg-MANWIT-6"></span></div></div></li> <li id="ePARMTF" onclick="ee('click','PARMTF')" onmouseover="ee('over','PARMTF')" onmouseout="ee('out','PARMTF')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/PARMTF"><span class="articleTitle recTitle">More trouble for regular probabilitites.</span></a><a class="discreet" title="View other works by Matthew W. Parker" href="/web/20230531191753/https://philpapers.org/s/Matthew W.%20Parker"><span class="name">Matthew W. Parker</span></a> - <span class="pubYear">2012</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">In standard probability theory, probability zero is not the same as impossibility. But many have suggested that only impossible events should have probability zero. This can be arranged if we allow infinitesimal probabilities, but infinitesimals do not solve all of the problems. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable<span id="PARMTF-absexp"> (<span class="ll" onclick="$("PARMTF-abstract2").show();$("PARMTF-absexp").hide()">...</span>)</span><span id="PARMTF-abstract2" style="display:none"> symmetry principles. Moreover, the examples here are immune to the objections against Williamson’s infinite coin flips. (<span class="ll" onclick="$("PARMTF-abstract2").hide();$("PARMTF-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-PARMTF"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/chance-and-objective-probability-misc" rel="section">Chance and Objective Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/indifference-principles" rel="section">Indifference Principles</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/prior-probabilities" rel="section">Prior Probabilities</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-principles-misc" rel="section">Probabilistic Principles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','PARMTF')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=PARMTF&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FPARMTF.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/PARMTF">(4 more)</a>   <div id="la-PARMTF" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('PARMTF')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-PARMTF" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('PARMTF','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/PARMTF"><i class="fa fa-share-alt"></i> 2 citations</a>   <span class="eMsg" id="msg-PARMTF"></span></div></div></li> <li id="eSERNIA-4" onclick="ee('click','SERNIA-4')" onmouseover="ee('over','SERNIA-4')" onmouseout="ee('out','SERNIA-4')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/SERNIA-4"><span class="articleTitle pub_name recTitle">Numerical infinities and infinitesimals in optimization.</span></a><a class="discreet" title="View other works by Yaroslav D. Sergeyev" href="/web/20230531191753/https://philpapers.org/s/Yaroslav D.%20Sergeyev"><span class="name">Yaroslav D. Sergeyev</span></a> & <a class="discreet" title="View other works by Renato De Leone" href="/web/20230531191753/https://philpapers.org/s/Renato%20De Leone"><span class="name">Renato De Leone</span></a> - <span class="pubYear">2022</span> - <span class="pubInfo"> 93413 Cham, Germania: Springer.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">From the Publisher: -/- This book presents a new powerful supercomputing paradigm introduced by Yaroslav D. Sergeyev -/- It gives a friendly introduction to the paradigm and proposes a broad panorama of a successful usage of numerical infinities -/- The volume covers software implementations of the Infinity Computer -/- Abstract -/- This book provides a friendly introduction to the paradigm and proposes a broad panorama of killing applications of the Infinity Computer in optimization: radically new numerical algorithms, great theoretical insights,<span id="SERNIA-4-absexp"> (<span class="ll" onclick="$("SERNIA-4-abstract2").show();$("SERNIA-4-absexp").hide()">...</span>)</span><span id="SERNIA-4-abstract2" style="display:none"> efficient software implementations, and interesting practical case studies. This is the first book presenting to the readers interested in optimization the advantages of a recently introduced supercomputing paradigm that allows to numerically work with different infinities and infinitesimals on the Infinity Computer patented in several countries. One of the editors of the book is the creator of the Infinity Computer, and another editor was the first who has started to use it in optimization. Their results were awarded by numerous scientific prizes. This engaging book opens new horizons for researchers, engineers, professors, and students with interests in supercomputing paradigms, optimization, decision making, game theory, and foundations of mathematics and computer science. -/- “Mathematicians have never been comfortable handling infinities… But an entirely new type of mathematics looks set to by-pass the problem… Today, Yaroslav Sergeyev, a mathematician at the University of Calabria in Italy solves this problem… ” -/- MIT Technology Review -/- “These ideas and future hardware prototypes may be productive in all fields of science where infinite and infinitesimal numbers (derivatives, integrals, series, fractals) are used.” A. Adamatzky, Editor-in-Chief of the International Journal of Unconventional Computing. -/- “I am sure that the new approach … will have a very deep impact both on Mathematics and Computer Science.” D. Trigiante, Computational Management Science. -/- “Within the grossone framework, it becomes feasible to deal computationally with infinite quantities, in a way that is both new (in the sense that previously intractable problems become amenable to computation) and natural”. R. Gangle, G. Caterina, F. Tohmé, Soft Computing. -/- “The computational features offered by the Infinity Computer allow us to dynamically change the accuracy of representation and floating-point operations during the flow of a computation. When suitably implemented, this possibility turns out to be particularly advantageous when solving ill-conditioned problems. In fact, compared with a standard multi-precision arithmetic, here the accuracy is improved only when needed, thus not affecting that much the overall computational effort.” P. Amodio, L. Brugnano, F. Iavernaro & F. Mazzia, Soft Computing. (<span class="ll" onclick="$("SERNIA-4-abstract2").hide();$("SERNIA-4-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-SERNIA-4"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/mathematical-cognition" rel="section">Mathematical Cognition</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/paradoxes" rel="section">Paradoxes</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/logic-and-philosophy-of-logic" rel="section">Logic and Philosophy of Logic</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-infinite" rel="section">The Infinite</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-nature-of-sets" rel="section">The Nature of Sets</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> </div><div class="options"><div class="affiliateLinks"><span class="price_new bargain"><a class="price_new bargain" target="_blank" rel="nofollow" href="https://web.archive.org/web/20230531191753/https://www.amazon.com/dp/3030936414?tag=philp02-20&linkCode=osi&th=1&psc=1">$134.88 new</a></span>   <a href="https://web.archive.org/web/20230531191753/https://www.amazon.com/dp/3030936414?tag=philp02-20&linkCode=osi&th=1&psc=1"> View on Amazon.com</a></div><span title="Remove from this list" class="ll" onclick="removeFromList('5874','SERNIA-4')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=SERNIA-4&proxyId=&u=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-030-93642-6"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/SERNIA-4">(2 more)</a>   <div id="la-SERNIA-4" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('SERNIA-4')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-SERNIA-4" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('SERNIA-4','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <span class="eMsg" id="msg-SERNIA-4"></span></div></div></li> <li id="eHANACO-12" onclick="ee('click','HANACO-12')" onmouseover="ee('over','HANACO-12')" onmouseout="ee('out','HANACO-12')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/HANACO-12"><span class="articleTitle recTitle">A Characterization of Probability-based Dichotomous Belief Revision.</span></a><a class="discreet" title="View other works by Sven Ove Hansson" href="/web/20230531191753/https://philpapers.org/s/Sven Ove%20Hansson"><span class="name">Sven Ove Hansson</span></a> - <span class="pubYear">2021</span> - <span class="pubInfo"> <i class="pubName">Studia Logica</i> 110 (2):511-543.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">This article investigates the properties of multistate top revision, a dichotomous model of belief revision that is based on an underlying model of probability revision. A proposition is included in the belief set if and only if its probability is either 1 or infinitesimally close to 1. Infinitesimal probabilities are used to keep track of propositions that are currently considered to have negligible probability, so that they are available if future information makes them more plausible. Multistate top revision satisfies a<span id="HANACO-12-absexp"> (<span class="ll" onclick="$("HANACO-12-abstract2").show();$("HANACO-12-absexp").hide()">...</span>)</span><span id="HANACO-12-abstract2" style="display:none"> slightly modified version of the set of basic and supplementary AGM postulates, except the inclusion and success postulates. This result shows that hyperreal probabilities can provide us with efficient tools for overcoming the well known difficulties in combining dichotomous and probabilistic models of belief change. (<span class="ll" onclick="$("HANACO-12-abstract2").hide();$("HANACO-12-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-HANACO-12"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/agm-belief-revision-theory" rel="section">AGM Belief Revision Theory</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/epistemology" rel="section">Epistemology</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','HANACO-12')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=HANACO-12&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1007%2Fs11225-021-09961-2"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/HANACO-12">(4 more)</a>   <div id="la-HANACO-12" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('HANACO-12')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-HANACO-12" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('HANACO-12','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/HANACO-12"><i class="fa fa-share-alt"></i> 1 citation</a>   <span class="eMsg" id="msg-HANACO-12"></span></div></div></li> <li id="ePARWRT" onclick="ee('click','PARWRT')" onmouseover="ee('over','PARWRT')" onmouseout="ee('out','PARWRT')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/PARWRT"><span class="articleTitle recTitle">Weintraub’s response to Williamson’s coin flip argument.</span></a><a class="discreet" title="View other works by Matthew W. Parker" href="/web/20230531191753/https://philpapers.org/s/Matthew W.%20Parker"><span class="name">Matthew W. Parker</span></a> - <span class="pubYear">2021</span> - <span class="pubInfo"> <i class="pubName">European Journal for Philosophy of Science</i> 11 (3):1-21.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">A probability distribution is regular if it does not assign probability zero to any possible event. Williamson argued that we should not require probabilities to be regular, for if we do, certain “isomorphic” physical events must have different probabilities, which is implausible. His remarks suggest an assumption that chances are determined by intrinsic, qualitative circumstances. Weintraub responds that Williamson’s coin flip events differ in their inclusion relations to each other, or the inclusion relations between their times, and this can account<span id="PARWRT-absexp"> (<span class="ll" onclick="$("PARWRT-abstract2").show();$("PARWRT-absexp").hide()">...</span>)</span><span id="PARWRT-abstract2" style="display:none"> for their differences in probability. Haverkamp and Schulz rebut Weintraub, but their rebuttal fails because the events in their example are even less symmetric than Williamson’s. However, Weintraub’s argument also fails, for it ignores the distinction between intrinsic, qualitative differences and relations of time and bare identity. Weintraub could rescue her argument by claiming that the events differ in duration, under a non-standard and problematic conception of duration. However, we can modify Williamson’s example with Special Relativity so that there is no absolute inclusion relation between the times, and neither event has longer duration except relative to certain reference frames. Hence, Weintraub’s responses do not apply unless chance is observer-relative, which is also problematic. Finally, another symmetry argument defeats even the appeal to frame-dependent durations, for there the events have the same finite duration and are entirely disjoint, as are their respective times and places. (<span class="ll" onclick="$("PARWRT-abstract2").hide();$("PARWRT-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-PARWRT"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/axioms-of-probability" rel="section">Axioms of Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/chance-and-objective-probability-misc" rel="section">Chance and Objective Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/degrees-of-belief" rel="section">Degrees of Belief</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-principles-misc" rel="section">Probabilistic Principles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','PARWRT')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=PARWRT&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1007%2Fs13194-021-00389-y"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/PARWRT">(3 more)</a>   <div id="la-PARWRT" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('PARWRT')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-PARWRT" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('PARWRT','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <span class="eMsg" id="msg-PARWRT"></span></div></div></li> <li id="eEDGHCP-2" onclick="ee('click','EDGHCP-2')" onmouseover="ee('over','EDGHCP-2')" onmouseout="ee('out','EDGHCP-2')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/EDGHCP-2"><span class="articleTitle recTitle">Hermann Cohen’s Principle of the Infinitesimal Method: A Defense.</span></a><a class="discreet" title="View other works by Scott Edgar" href="/web/20230531191753/https://philpapers.org/s/Scott%20Edgar"><span class="name">Scott Edgar</span></a> - <span class="pubYear">2020</span> - <span class="pubInfo"> <i class="pubName">Hopos: The Journal of the International Society for the History of Philosophy of Science</i> 10 (2):440-470.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">In Bertrand Russell's 1903 Principles of Mathematics, he offers an apparently devastating criticism of the neo-Kantian Hermann Cohen's Principle of the Infinitesimal Method and its History (PIM). Russell's criticism is motivated by his concern that Cohen's account of the foundations of calculus saddles mathematics with the paradoxes of the infinitesimal and continuum, and thus threatens the very idea of mathematical truth. This paper defends Cohen against that objection of Russell's, and argues that properly understood, Cohen's views of limits and infinitesimals<span id="EDGHCP-2-absexp"> (<span class="ll" onclick="$("EDGHCP-2-abstract2").show();$("EDGHCP-2-absexp").hide()">...</span>)</span><span id="EDGHCP-2-abstract2" style="display:none"> do not entail the paradoxes of the infinitesimal and continuum. Essential to that defense is an interpretation, developed in the paper, of Cohen's positions in the PIM as deeply rationalist. The interest in developing this interpretation is not just that it reveals how Cohen's views in the PIM avoid the paradoxes of the infinitesimal and continuum. It also reveals some of what is at stake, both historically and philosophically, in Russell's criticism of Cohen. (<span class="ll" onclick="$("EDGHCP-2-abstract2").hide();$("EDGHCP-2-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-EDGHCP-2"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/19th-century-german-philosophy" rel="section">19th Century German Philosophy</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/19th-century-philosophy" rel="section">19th Century Philosophy</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/analysis" rel="section">Analysis</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/history-of-mathematics" rel="section">History of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/history-philosophy-of-mathematics" rel="section">History: Philosophy of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/kant-philosophy-of-mathematics" rel="section">Kant: Philosophy of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/17th18th-century-philosophy" rel="section">17th/18th Century Philosophy</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/mathematical-truth" rel="section">Mathematical Truth</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-infinite" rel="section">The Infinite</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','EDGHCP-2')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=EDGHCP-2&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FEDGHCP-2.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/EDGHCP-2">(3 more)</a>   <div id="la-EDGHCP-2" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('EDGHCP-2')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-EDGHCP-2" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('EDGHCP-2','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/EDGHCP-2"><i class="fa fa-share-alt"></i> 2 citations</a>   <span class="eMsg" id="msg-EDGHCP-2"></span></div></div></li> <li id="eMCCUWA-2" onclick="ee('click','MCCUWA-2')" onmouseover="ee('over','MCCUWA-2')" onmouseout="ee('out','MCCUWA-2')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/MCCUWA-2"><span class="articleTitle recTitle">Utilitarianism with and without expected utility.</span></a><a class="discreet" title="View other works by David McCarthy" href="/web/20230531191753/https://philpapers.org/s/David%20McCarthy"><span class="name">David McCarthy</span></a>, <a class="discreet" title="View other works by Kalle Mikkola" href="/web/20230531191753/https://philpapers.org/s/Kalle%20Mikkola"><span class="name">Kalle Mikkola</span></a> & <a class="discreet" title="View other works by Joaquin Teruji Thomas" href="/web/20230531191753/https://philpapers.org/s/Joaquin Teruji%20Thomas"><span class="name">Joaquin Teruji Thomas</span></a> - <span class="pubYear">2020</span> - <span class="pubInfo"> <i class="pubName">Journal of Mathematical Economics</i> 87:77-113.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are<span id="MCCUWA-2-absexp"> (<span class="ll" onclick="$("MCCUWA-2-abstract2").show();$("MCCUWA-2-absexp").hide()">...</span>)</span><span id="MCCUWA-2-abstract2" style="display:none"> also consistent with the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expected utility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expected utility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi’s utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a ‘local expected utility’ condition popular in non-expected utility theory, then the social preorder has a ‘local expected total utility’ representation. Fourth, a wide range of non-expected utility theories nevertheless lead to social preorders of outcomes that have been seen as canonically egalitarian, such as rank-dependent social preorders. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity. (<span class="ll" onclick="$("MCCUWA-2-abstract2").hide();$("MCCUWA-2-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-MCCUWA-2"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/decision-theory-and-ethics" rel="section">Decision Theory and Ethics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-action" rel="section">Philosophy of Action</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/distributive-justice-misc" rel="section">Distributive Justice, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/social-and-political-philosophy" rel="section">Social and Political Philosophy</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/economics" rel="section">Economics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/social-sciences" rel="section">Social Sciences</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/egalitarianism-misc" rel="section">Egalitarianism, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/social-and-political-philosophy" rel="section">Social and Political Philosophy</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/incommensurability-of-value" rel="section">Incommensurability of Value</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/value-theory-miscellaneous" rel="section">Value Theory, Miscellaneous</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/population-ethics" rel="section">Population Ethics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/value-theory-miscellaneous" rel="section">Value Theory, Miscellaneous</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/utilitarianism-misc" rel="section">Utilitarianism, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/normative-ethics" rel="section">Normative Ethics</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','MCCUWA-2')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=MCCUWA-2&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FMCCUWA-2.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/MCCUWA-2">(2 more)</a>   <div id="la-MCCUWA-2" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('MCCUWA-2')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-MCCUWA-2" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('MCCUWA-2','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/MCCUWA-2"><i class="fa fa-share-alt"></i> 4 citations</a>   <span class="eMsg" id="msg-MCCUWA-2"></span></div></div></li> <li id="ePARSAA-10" onclick="ee('click','PARSAA-10')" onmouseover="ee('over','PARSAA-10')" onmouseout="ee('out','PARSAA-10')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/PARSAA-10"><span class="articleTitle recTitle">Symmetry arguments against regular probability: A reply to recent objections.</span></a><a class="discreet" title="View other works by Matthew W. Parker" href="/web/20230531191753/https://philpapers.org/s/Matthew W.%20Parker"><span class="name">Matthew W. Parker</span></a> - <span class="pubYear">2019</span> - <span class="pubInfo"> <i class="pubName">European Journal for Philosophy of Science</i> 9 (1):1-21.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">A probability distribution is regular if it does not assign probability zero to any possible event. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson and Benci et al. have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s “isomorphic” events are not in fact isomorphic, but Howson is speaking<span id="PARSAA-10-absexp"> (<span class="ll" onclick="$("PARSAA-10-abstract2").show();$("PARSAA-10-absexp").hide()">...</span>)</span><span id="PARSAA-10-abstract2" style="display:none"> of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (<span class="ll" onclick="$("PARSAA-10-abstract2").hide();$("PARSAA-10-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-PARSAA-10"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/chance-and-objective-probability-misc" rel="section">Chance and Objective Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-principles-misc" rel="section">Probabilistic Principles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probability-in-the-physical-sciences-misc" rel="section">Probability in the Physical Sciences, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/subjective-probability-misc" rel="section">Subjective Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','PARSAA-10')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=PARSAA-10&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FPARSAA-10.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/PARSAA-10">(3 more)</a>   <div id="la-PARSAA-10" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('PARSAA-10')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-PARSAA-10" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('PARSAA-10','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/PARSAA-10"><i class="fa fa-share-alt"></i> 6 citations</a>   <span class="eMsg" id="msg-PARSAA-10"></span></div></div></li> <li id="ePARSAA-8" onclick="ee('click','PARSAA-8')" onmouseover="ee('over','PARSAA-8')" onmouseout="ee('out','PARSAA-8')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/PARSAA-8"><span class="articleTitle recTitle">Symmetry arguments against regular probability: A reply to recent objections.</span></a><a class="discreet" title="View other works by Matthew W. Parker" href="/web/20230531191753/https://philpapers.org/s/Matthew W.%20Parker"><span class="name">Matthew W. Parker</span></a> - <span class="pubYear">2018</span> - <span class="pubInfo"> <i class="pubName">European Journal for Philosophy of Science</i> 9 (1):8.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">A probability distribution is regular if no possible event is assigned probability zero. While some hold that probabilities should always be regular, three counter-arguments have been posed based on examples where, if regularity holds, then perfectly similar events must have different probabilities. Howson (2017) and Benci et al. (2016) have raised technical objections to these symmetry arguments, but we see here that their objections fail. Howson says that Williamson’s (2007) “isomorphic” events are not in fact isomorphic, but Howson is speaking<span id="PARSAA-8-absexp"> (<span class="ll" onclick="$("PARSAA-8-abstract2").show();$("PARSAA-8-absexp").hide()">...</span>)</span><span id="PARSAA-8-abstract2" style="display:none"> of set-theoretic representations of events in a probability model. While those sets are not isomorphic, Williamson’s physical events are, in the relevant sense. Benci et al. claim that all three arguments rest on a conflation of different models, but they do not. They are founded on the premise that similar events should have the same probability in the same model, or in one case, on the assumption that a single rotation-invariant distribution is possible. Having failed to refute the symmetry arguments on such technical grounds, one could deny their implicit premises, which is a heavy cost, or adopt varying degrees of instrumentalism or pluralism about regularity, but that would not serve the project of accurately modelling chances. (<span class="ll" onclick="$("PARSAA-8-abstract2").hide();$("PARSAA-8-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-PARSAA-8"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-principles-misc" rel="section">Probabilistic Principles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','PARSAA-8')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=PARSAA-8&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FPARSAA-8.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/PARSAA-8">(5 more)</a>   <div id="la-PARSAA-8" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('PARSAA-8')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-PARSAA-8" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('PARSAA-8','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/PARSAA-8"><i class="fa fa-share-alt"></i> 7 citations</a>   <span class="eMsg" id="msg-PARSAA-8"></span></div></div></li> <li id="eWHIRWP" onclick="ee('click','WHIRWP')" onmouseover="ee('over','WHIRWP')" onmouseout="ee('out','WHIRWP')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/WHIRWP"><span class="articleTitle recTitle">Reasoning with Plenitude.</span></a><a class="discreet" title="View other works by Roger White" href="/web/20230531191753/https://philpapers.org/s/Roger%20White"><span class="name">Roger White</span></a> - <span class="pubYear">2018</span> - <span class="pubInfo"> In Matthew A. Benton, John Hawthorne & Dani Rabinowitz (eds.), <i><a href="https://web.archive.org/web/20230531191753/https://philpapers.org/rec/BENKBA-3">Knowledge, Belief, and God: New Insights in Religious Epistemology</a></i>. Oxford: Oxford University Press. pp. 169-179.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="catsCon" id="ecats-con-WHIRWP"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/epistemology-of-disagreement" rel="section">Epistemology of Disagreement</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/epistemology" rel="section">Epistemology</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/multiple-universes" rel="section">Multiple Universes</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-physical-science" rel="section">Philosophy of Physical Science</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/religious-diversity-misc" rel="section">Religious Diversity, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-religion" rel="section">Philosophy of Religion</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','WHIRWP')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=WHIRWP&proxyId=&u=https%3A%2F%2Fglobal.oup.com%2Facademic%2Fproduct%2Fknowledge-belief-and-god-9780198798705"><i class="fa fa-download"></i> Direct download</a>   <div id="la-WHIRWP" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('WHIRWP')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-WHIRWP" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('WHIRWP','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/WHIRWP"><i class="fa fa-share-alt"></i> 2 citations</a>   <span class="eMsg" id="msg-WHIRWP"></span></div></div></li> <li id="eHOFIC" onclick="ee('click','HOFIC')" onmouseover="ee('over','HOFIC')" onmouseout="ee('out','HOFIC')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/HOFIC"><span class="articleTitle recTitle">Infinitesimal Chances.</span></a><a class="discreet" title="View other works by Thomas Hofweber" href="/web/20230531191753/https://philpapers.org/s/Thomas%20Hofweber"><span class="name">Thomas Hofweber</span></a> - <span class="pubYear">2014</span> - <span class="pubInfo"> <i class="pubName">Philosophers' Imprint</i> 14.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">It is natural to think that questions in the metaphysics of chance are independent of the mathematical representation of chance in probability theory. After all, chance is a feature of events that comes in degrees and the mathematical representation of chance concerns these degrees but leaves the nature of chance open. The mathematical representation of chance could thus, un-controversially, be taken to be what it is commonly taken to be: a probability measure satisfying Kolmogorov’s axioms. The metaphysical questions about chance<span id="HOFIC-absexp"> (<span class="ll" onclick="$("HOFIC-abstract2").show();$("HOFIC-absexp").hide()">...</span>)</span><span id="HOFIC-abstract2" style="display:none"> seem to be left open by all this. I argue that this is a mistake. The employment of real numbers as measures of chance in standard probability theory brings with it commitments in the metaphysics of (objective) chance that are not only substantial but also mistaken. To measure chance properly we need to employ extensions of the real numbers that contain infinitesimals: positive numbers that are infinitely small. But simply using infinitesimals alone is not enough, as a number of arguments show. Instead we need to put three ideas together: infinitesimals, the non-locality of chance and flexibility in measurement. Only those three together give us a coherent picture of chance and its mathematical representation. (<span class="ll" onclick="$("HOFIC-abstract2").hide();$("HOFIC-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-HOFIC"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/chance-and-objective-probability-misc" rel="section">Chance and Objective Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/mathematics-of-probability-misc" rel="section">Mathematics of Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','HOFIC')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=HOFIC&proxyId=&u=http%3A%2F%2Fhdl.handle.net%2F2027%2Fspo.3521354.0014.002"><i class="fa fa-download"></i> Direct download</a>   <div id="la-HOFIC" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('HOFIC')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-HOFIC" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('HOFIC','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/HOFIC"><i class="fa fa-share-alt"></i> 17 citations</a>   <span class="eMsg" id="msg-HOFIC"></span></div></div></li> <li id="eKREIOF" onclick="ee('click','KREIOF')" onmouseover="ee('over','KREIOF')" onmouseout="ee('out','KREIOF')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/KREIOF"><span class="articleTitle recTitle">Indeterminacy of fair infinite lotteries.</span></a><a class="discreet" title="View other works by Philip Kremer" href="/web/20230531191753/https://philpapers.org/s/Philip%20Kremer"><span class="name">Philip Kremer</span></a> - <span class="pubYear">2014</span> - <span class="pubInfo"> <i class="pubName">Synthese</i> 191 (8):1757-1760.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">In ‘Fair Infinite Lotteries’ (FIL), Wenmackers and Horsten use non-standard analysis to construct a family of nicely-behaved hyperrational-valued probability measures on sets of natural numbers. Each probability measure in FIL is determined by a free ultrafilter on the natural numbers: distinct free ultrafilters determine distinct probability measures. The authors reply to a worry about a consequent ‘arbitrariness’ by remarking, “A different choice of free ultrafilter produces a different ... probability function with the same standard part but infinitesimal differences.” They illustrate<span id="KREIOF-absexp"> (<span class="ll" onclick="$("KREIOF-abstract2").show();$("KREIOF-absexp").hide()">...</span>)</span><span id="KREIOF-abstract2" style="display:none"> this remark with the example of the sets of odd and even numbers. Depending on the ultrafilter, either each of these sets has probability 1/2, or the set of odd numbers has a probability infinitesimally higher than 1/2 and the set of even numbers infinitesimally lower. The point of the current paper is simply that the amount of indeterminacy is much greater than acknowledged in FIL: there are sets of natural numbers whose probability is far more indeterminate than that of the set of odd or the set of even numbers. (<span class="ll" onclick="$("KREIOF-abstract2").hide();$("KREIOF-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-KREIOF"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/applications-of-probability" rel="section">Applications of Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','KREIOF')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=KREIOF&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1007%2Fs11229-013-0364-3"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/KREIOF">(3 more)</a>   <div id="la-KREIOF" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('KREIOF')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-KREIOF" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('KREIOF','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/KREIOF"><i class="fa fa-share-alt"></i> 6 citations</a>   <span class="eMsg" id="msg-KREIOF"></span></div></div></li> <li id="eLPEBEO" onclick="ee('click','LPEBEO')" onmouseover="ee('over','LPEBEO')" onmouseout="ee('out','LPEBEO')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/LPEBEO"><span class="articleTitle recTitle">Berkeley: El origen de la crítica a los infinitesimales / Berkeley: The Origin of his Critics to Infinitesimals.</span></a><a class="discreet" title="View other works by Alberto Luis López" href="/web/20230531191753/https://philpapers.org/s/Alberto Luis%20López"><span class="name">Alberto Luis López</span></a> - <span class="pubYear">2014</span> - <span class="pubInfo"> <i class="pubName">Cuadernos Salmantinos de Filosofía</i> 41 (1):195-217.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">BERKELEY: THE ORIGIN OF CRITICISM OF THE INFINITESIMALS Abstract: In this paper I propose a new reading of a little known George Berkeley´s work Of Infinites. Hitherto, the work has been studied partially, or emphasizing only the mathematical contributions, downplaying the philosophical aspects, or minimizing mathematical issues taking into account only the incipient immaterialism. Both readings have been pernicious for the correct comprehension of the work and that has brought as a result that will follow underestimated its importance, and therefore<span id="LPEBEO-absexp"> (<span class="ll" onclick="$("LPEBEO-abstract2").show();$("LPEBEO-absexp").hide()">...</span>)</span><span id="LPEBEO-abstract2" style="display:none"> will not study as should be. Against traditional readings I make one that stand out both philosophical and mathematical aspects, with the purpose to show that richness and complexity of the work deserve that it has an special place within Berkeley´s works. (<span class="ll" onclick="$("LPEBEO-abstract2").hide();$("LPEBEO-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-LPEBEO"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/berkeley-philosophy-of-science" rel="section">Berkeley: Philosophy of Science</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/17th18th-century-philosophy" rel="section">17th/18th Century Philosophy</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/history-of-mathematics" rel="section">History of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','LPEBEO')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=LPEBEO&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FLPEBEO.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/LPEBEO">(2 more)</a>   <div id="la-LPEBEO" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('LPEBEO')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-LPEBEO" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('LPEBEO','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <span class="eMsg" id="msg-LPEBEO"></span></div></div></li> <li id="ePRUIAT-2" onclick="ee('click','PRUIAT-2')" onmouseover="ee('over','PRUIAT-2')" onmouseout="ee('out','PRUIAT-2')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/PRUIAT-2"><span class="articleTitle recTitle">Infinitesimals are too small for countably infinite fair lotteries.</span></a><a class="discreet" title="View other works by Alexander R. Pruss" href="/web/20230531191753/https://philpapers.org/s/Alexander R.%20Pruss"><span class="name">Alexander R. Pruss</span></a> - <span class="pubYear">2014</span> - <span class="pubInfo"> <i class="pubName">Synthese</i> 191 (6):1051-1057.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">We show that infinitesimal probabilities are much too small for modeling the individual outcome of a countably infinite fair lottery. </div><div class="catsCon" id="ecats-con-PRUIAT-2"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/areas-of-mathematics" rel="section">Areas of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','PRUIAT-2')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=PRUIAT-2&proxyId=&u=http%3A%2F%2Flink.springer.com%2Fcontent%2Fpdf%2F10.1007%252Fs11229-013-0307-z.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/PRUIAT-2">(5 more)</a>   <div id="la-PRUIAT-2" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('PRUIAT-2')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-PRUIAT-2" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('PRUIAT-2','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/PRUIAT-2"><i class="fa fa-share-alt"></i> 20 citations</a>   <span class="eMsg" id="msg-PRUIAT-2"></span></div></div></li> <li id="eBENNP-2" onclick="ee('click','BENNP-2')" onmouseover="ee('over','BENNP-2')" onmouseout="ee('out','BENNP-2')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/BENNP-2"><span class="articleTitle recTitle">Non-Archimedean Probability.</span></a><a class="discreet" title="View other works by Vieri Benci" href="/web/20230531191753/https://philpapers.org/s/Vieri%20Benci"><span class="name">Vieri Benci</span></a>, <a class="discreet" title="View other works by Leon Horsten" href="/web/20230531191753/https://philpapers.org/s/Leon%20Horsten"><span class="name">Leon Horsten</span></a> & <a class="discreet" title="View other works by Sylvia Wenmackers" href="/web/20230531191753/https://philpapers.org/s/Sylvia%20Wenmackers"><span class="name">Sylvia Wenmackers</span></a> - <span class="pubYear">2013</span> - <span class="pubInfo"> <i class="pubName">Milan Journal of Mathematics</i> 81 (1):121-151.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">We propose an alternative approach to probability theory closely related to the framework of numerosity theory: non-Archimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a non-Archimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by<span id="BENNP-2-absexp"> (<span class="ll" onclick="$("BENNP-2-abstract2").show();$("BENNP-2-absexp").hide()">...</span>)</span><span id="BENNP-2-abstract2" style="display:none"> a different type of infinite additivity. (<span class="ll" onclick="$("BENNP-2-abstract2").hide();$("BENNP-2-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-BENNP-2"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/axioms-of-probability" rel="section">Axioms of Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/mathematics" rel="section">Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/formal-sciences" rel="section">Formal Sciences</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','BENNP-2')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=BENNP-2&proxyId=&u=http%3A%2F%2Farxiv.org%2Fabs%2F1106.1524"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/BENNP-2">(2 more)</a>   <div id="la-BENNP-2" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('BENNP-2')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-BENNP-2" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('BENNP-2','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/BENNP-2"><i class="fa fa-share-alt"></i> 34 citations</a>   <span class="eMsg" id="msg-BENNP-2"></span></div></div></li> <li id="eWENFIL" onclick="ee('click','WENFIL')" onmouseover="ee('over','WENFIL')" onmouseout="ee('out','WENFIL')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/WENFIL"><span class="articleTitle recTitle">Fair infinite lotteries.</span></a><a class="discreet" title="View other works by Sylvia Wenmackers" href="/web/20230531191753/https://philpapers.org/s/Sylvia%20Wenmackers"><span class="name">Sylvia Wenmackers</span></a> & <a class="discreet" title="View other works by Leon Horsten" href="/web/20230531191753/https://philpapers.org/s/Leon%20Horsten"><span class="name">Leon Horsten</span></a> - <span class="pubYear">2013</span> - <span class="pubInfo"> <i class="pubName">Synthese</i> 190 (1):37-61.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from non-standard analysis are brought to bear on the problem. </div><div class="catsCon" id="ecats-con-WENFIL"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/nonstandard-axiomatizations" rel="section">Nonstandard Axiomatizations</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-puzzles-misc" rel="section">Probabilistic Puzzles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','WENFIL')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=WENFIL&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FWENFIL.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/WENFIL">(9 more)</a>   <div id="la-WENFIL" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('WENFIL')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-WENFIL" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('WENFIL','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/WENFIL"><i class="fa fa-share-alt"></i> 38 citations</a>   <span class="eMsg" id="msg-WENFIL"></span></div></div></li> <li id="eBENAFN" onclick="ee('click','BENAFN')" onmouseover="ee('over','BENAFN')" onmouseout="ee('out','BENAFN')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/BENAFN"><span class="articleTitle recTitle">Axioms for Non-Archimedean Probability (NAP).</span></a><a class="discreet" title="View other works by Vieri Benci" href="/web/20230531191753/https://philpapers.org/s/Vieri%20Benci"><span class="name">Vieri Benci</span></a>, <a class="discreet" title="View other works by Leon Horsten" href="/web/20230531191753/https://philpapers.org/s/Leon%20Horsten"><span class="name">Leon Horsten</span></a> & <a class="discreet" title="View other works by Sylvia Wenmackers" href="/web/20230531191753/https://philpapers.org/s/Sylvia%20Wenmackers"><span class="name">Sylvia Wenmackers</span></a> - <span class="pubYear">2012</span> - <span class="pubInfo"> In De Vuyst J. & Demey L. (eds.), <i><a href="https://web.archive.org/web/20230531191753/https://philpapers.org/rec/JFDF">Future Directions for Logic; Proceedings of PhDs in Logic III - Vol. 2 of IfColog Proceedings</a></i>. College Publications.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">In this contribution, we focus on probabilistic problems with a denumerably or non-denumerably infinite number of possible outcomes. Kolmogorov (1933) provided an axiomatic basis for probability theory, presented as a part of measure theory, which is a branch of standard analysis or calculus. Since standard analysis does not allow for non-Archimedean quantities (i.e. infinitesimals), we may call Kolmogorov's approach "Archimedean probability theory". We show that allowing non-Archimedean probability values may have considerable epistemological advantages in the infinite case. The current paper<span id="BENAFN-absexp"> (<span class="ll" onclick="$("BENAFN-abstract2").show();$("BENAFN-absexp").hide()">...</span>)</span><span id="BENAFN-abstract2" style="display:none"> focuses on the motivation for our new axiomatization. (<span class="ll" onclick="$("BENAFN-abstract2").hide();$("BENAFN-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-BENAFN"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/axioms-of-probability" rel="section">Axioms of Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-infinite" rel="section">The Infinite</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','BENAFN')"><i class="fa fa-times"></i> Remove from this list</span>   <div id="la-BENAFN" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('BENAFN')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-BENAFN" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('BENAFN','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/BENAFN"><i class="fa fa-share-alt"></i> 1 citation</a>   <span class="eMsg" id="msg-BENAFN"></span></div></div></li> <li id="eHAVANO" onclick="ee('click','HAVANO')" onmouseover="ee('over','HAVANO')" onmouseout="ee('out','HAVANO')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/HAVANO"><span class="articleTitle recTitle">A Note on Comparative Probability.</span></a><a class="discreet" title="View other works by Nick Haverkamp" href="/web/20230531191753/https://philpapers.org/s/Nick%20Haverkamp"><span class="name">Nick Haverkamp</span></a> & <a class="discreet" title="View other works by Moritz Schulz" href="/web/20230531191753/https://philpapers.org/s/Moritz%20Schulz"><span class="name">Moritz Schulz</span></a> - <span class="pubYear">2012</span> - <span class="pubInfo"> <i class="pubName">Erkenntnis</i> 76 (3):395-402.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">A possible event always seems to be more probable than an impossible event. Although this constraint, usually alluded to as regularity , is prima facie very attractive, it cannot hold for standard probabilities. Moreover, in a recent paper Timothy Williamson has challenged even the idea that regularity can be integrated into a comparative conception of probability by showing that the standard comparative axioms conflict with certain cases if regularity is assumed. In this note, we suggest that there is a natural<span id="HAVANO-absexp"> (<span class="ll" onclick="$("HAVANO-abstract2").show();$("HAVANO-absexp").hide()">...</span>)</span><span id="HAVANO-abstract2" style="display:none"> weakening of the standard comparative axioms. It is shown that these axioms are consistent both with the regularity condition and with the essential feature of Williamson’s example. (<span class="ll" onclick="$("HAVANO-abstract2").hide();$("HAVANO-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-HAVANO"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/axioms-of-probability" rel="section">Axioms of Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/interpretation-of-probability-misc" rel="section">Interpretation of Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-frameworks" rel="section">Probabilistic Frameworks</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-principles-misc" rel="section">Probabilistic Principles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','HAVANO')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=HAVANO&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1007%2Fs10670-011-9307-x"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/HAVANO">(5 more)</a>   <div id="la-HAVANO" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('HAVANO')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-HAVANO" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('HAVANO','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/HAVANO"><i class="fa fa-share-alt"></i> 4 citations</a>   <span class="eMsg" id="msg-HAVANO"></span></div></div></li> <li id="ePRUILP" onclick="ee('click','PRUILP')" onmouseover="ee('over','PRUILP')" onmouseout="ee('out','PRUILP')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/PRUILP"><span class="articleTitle recTitle">Infinite Lotteries, Perfectly Thin Darts and Infinitesimals.</span></a><a class="discreet" title="View other works by Alexander R. Pruss" href="/web/20230531191753/https://philpapers.org/s/Alexander R.%20Pruss"><span class="name">Alexander R. Pruss</span></a> - <span class="pubYear">2012</span> - <span class="pubInfo"> <i class="pubName">Thought: A Journal of Philosophy</i> 1 (2):81-89.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">One of the problems that Bayesian regularity, the thesis that all contingent propositions should be given probabilities strictly between zero and one, faces is the possibility of random processes that randomly and uniformly choose a number between zero and one. According to classical probability theory, the probability that such a process picks a particular number in the range is zero, but of course any number in the range can indeed be picked. There is a solution to this particular problem on<span id="PRUILP-absexp"> (<span class="ll" onclick="$("PRUILP-abstract2").show();$("PRUILP-absexp").hide()">...</span>)</span><span id="PRUILP-abstract2" style="display:none"> the books: a measure that assigns the same infinitesimal probability to each number between zero and one. I will show that such a measure, while mathematically interesting, is pathological for use in confirmation theory, for the same reason that a measure that assigns an infinitesimal probability to each possible outcome in a countably infinite lottery is pathological. The pathology is that one can force someone to assign a probability within an infinitesimal of one to an unlikely event. (<span class="ll" onclick="$("PRUILP-abstract2").hide();$("PRUILP-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-PRUILP"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/mathematics-of-probability-misc" rel="section">Mathematics of Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-puzzles-misc" rel="section">Probabilistic Puzzles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','PRUILP')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=PRUILP&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1002%2Ftht3.13"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/PRUILP">(3 more)</a>   <div id="la-PRUILP" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('PRUILP')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-PRUILP" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('PRUILP','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/PRUILP"><i class="fa fa-share-alt"></i> 21 citations</a>   <span class="eMsg" id="msg-PRUILP"></span></div></div></li> <li id="eWENUAI" onclick="ee('click','WENUAI')" onmouseover="ee('over','WENUAI')" onmouseout="ee('out','WENUAI')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/WENUAI"><span class="articleTitle recTitle">Ultralarge and infinite lotteries.</span></a><a class="discreet" title="View other works by Sylvia Wenmackers" href="/web/20230531191753/https://philpapers.org/s/Sylvia%20Wenmackers"><span class="name">Sylvia Wenmackers</span></a> - <span class="pubYear">2012</span> - <span class="pubInfo"> In B. Van Kerkhove, T. Libert, G. Vanpaemel & P. Marage (eds.), <i><a href="https://web.archive.org/web/20230531191753/https://philpapers.org/rec/VANLPA-3">Logic, Philosophy and History of Science in Belgium II (Proceedings of the Young Researchers Days 2010)</a></i>. Koninklijke Vlaamse Academie van België voor Wetenschappen en Kunsten.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">By exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. We solve the 'adding problems' that occur in these two contexts using a similar strategy, based on non-standard analysis. </div><div class="catsCon" id="ecats-con-WENUAI"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-puzzles-misc" rel="section">Probabilistic Puzzles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','WENUAI')"><i class="fa fa-times"></i> Remove from this list</span>   <div id="la-WENUAI" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('WENUAI')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-WENUAI" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('WENUAI','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/WENUAI"><i class="fa fa-share-alt"></i> 2 citations</a>   <span class="eMsg" id="msg-WENUAI"></span></div></div></li> <li id="eWENPOP" onclick="ee('click','WENPOP')" onmouseover="ee('over','WENPOP')" onmouseout="ee('out','WENPOP')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/WENPOP"><span class="articleTitle pub_name recTitle">Philosophy of Probability: Foundations, Epistemology, and Computation.</span></a><a class="discreet" title="View other works by Sylvia Wenmackers" href="/web/20230531191753/https://philpapers.org/s/Sylvia%20Wenmackers"><span class="name">Sylvia Wenmackers</span></a> - <span class="pubYear">2011</span> - <span class="pubInfo"> Dissertation, University of Groningen</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">This dissertation is a contribution to formal and computational philosophy. -/- In the first part, we show that by exploiting the parallels between large, yet finite lotteries on the one hand and countably infinite lotteries on the other, we gain insights in the foundations of probability theory as well as in epistemology. Case 1: Infinite lotteries. We discuss how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. The solution boils down to the introduction<span id="WENPOP-absexp"> (<span class="ll" onclick="$("WENPOP-abstract2").show();$("WENPOP-absexp").hide()">...</span>)</span><span id="WENPOP-abstract2" style="display:none"> of infinitesimal probability values, which can be achieved using non-standard analysis. Our solution can be generalized to uncountable sample spaces, giving rise to a Non-Archimedean Probability (NAP) theory. Case 2: Large but finite lotteries. We propose application of the language of relative analysis (a type of non-standard analysis) to formulate a new model for rational belief, called Stratified Belief. This contextualist model seems well-suited to deal with a concept of beliefs based on probabilities ‘sufficiently close to unity’. -/- The second part presents a case study in social epistemology. We model a group of agents who update their opinions by averaging the opinions of other agents. Our main goal is to calculate the probability for an agent to end up in an inconsistent belief state due to updating. To that end, an analytical expression is given and evaluated numerically, both exactly and using statistical sampling. The probability of ending up in an inconsistent belief state turns out to be always smaller than 2%. (<span class="ll" onclick="$("WENPOP-abstract2").hide();$("WENPOP-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-WENPOP"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/formal-epistemology" rel="section">Formal Epistemology</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/epistemology" rel="section">Epistemology</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/judgment-aggregation" rel="section">Judgment Aggregation</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/social-and-political-philosophy" rel="section">Social and Political Philosophy</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-infinite" rel="section">The Infinite</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','WENPOP')"><i class="fa fa-times"></i> Remove from this list</span>   <a class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=WENPOP&proxyId=&u=https%3A%2F%2Fphilpapers.org%2Farchive%2FWENPOP.pdf"><i class="fa fa-download"></i> Direct download</a>   <div id="la-WENPOP" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('WENPOP')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-WENPOP" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('WENPOP','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/WENPOP"><i class="fa fa-share-alt"></i> 5 citations</a>   <span class="eMsg" id="msg-WENPOP"></span></div></div></li> <li id="eSERNCA" onclick="ee('click','SERNCA')" onmouseover="ee('over','SERNCA')" onmouseout="ee('out','SERNCA')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/SERNCA"><span class="articleTitle recTitle">Numerical computations and mathematical modelling with infinite and infinitesimal numbers.</span></a><a class="discreet" title="View other works by Yaroslav Sergeyev" href="/web/20230531191753/https://philpapers.org/s/Yaroslav%20Sergeyev"><span class="name">Yaroslav Sergeyev</span></a> - <span class="pubYear">2009</span> - <span class="pubInfo"> <i class="pubName">Journal of Applied Mathematics and Computing</i> 29:177-195.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the non-standard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give possibilities to<span id="SERNCA-absexp"> (<span class="ll" onclick="$("SERNCA-abstract2").show();$("SERNCA-absexp").hide()">...</span>)</span><span id="SERNCA-abstract2" style="display:none"> execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given. (<span class="ll" onclick="$("SERNCA-abstract2").hide();$("SERNCA-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-SERNCA"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/computer-science" rel="section">Computer Science</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/formal-sciences" rel="section">Formal Sciences</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/mathematics" rel="section">Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/formal-sciences" rel="section">Formal Sciences</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-infinite" rel="section">The Infinite</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/theory-of-computation-misc" rel="section">Theory of Computation, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-computing-and-information" rel="section">Philosophy of Computing and Information</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','SERNCA')"><i class="fa fa-times"></i> Remove from this list</span>   <div id="la-SERNCA" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('SERNCA')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-SERNCA" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('SERNCA','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/SERNCA"><i class="fa fa-share-alt"></i> 2 citations</a>   <span class="eMsg" id="msg-SERNCA"></span></div></div></li> <li id="eWILHPI-4" onclick="ee('click','WILHPI-4')" onmouseover="ee('over','WILHPI-4')" onmouseout="ee('out','WILHPI-4')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/WILHPI-4"><span class="articleTitle recTitle">How probable is an infinite sequence of heads?</span></a><a class="discreet" title="View other works by Timothy Williamson" href="/web/20230531191753/https://philpapers.org/s/Timothy%20Williamson"><span class="name">Timothy Williamson</span></a> - <span class="pubYear">2007</span> - <span class="pubInfo"> <i class="pubName">Analysis</i> 67 (3):173-180.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">Isn't probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur. If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur. And so on for other<span id="WILHPI-4-absexp"> (<span class="ll" onclick="$("WILHPI-4-abstract2").show();$("WILHPI-4-absexp").hide()">...</span>)</span><span id="WILHPI-4-abstract2" style="display:none"> kinds of probability, such as evidential probability. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. (<span class="ll" onclick="$("WILHPI-4-abstract2").hide();$("WILHPI-4-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-WILHPI-4"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','WILHPI-4')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=WILHPI-4&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1093%2Fanalys%2F67.3.173"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/WILHPI-4">(4 more)</a>   <div id="la-WILHPI-4" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('WILHPI-4')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-WILHPI-4" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('WILHPI-4','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/WILHPI-4"><i class="fa fa-share-alt"></i> 80 citations</a>   <span class="eMsg" id="msg-WILHPI-4"></span></div></div></li> <li id="eOPPPPO-2" onclick="ee('click','OPPPPO-2')" onmouseover="ee('over','OPPPPO-2')" onmouseout="ee('out','OPPPPO-2')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/OPPPPO-2"><span class="articleTitle pub_name recTitle">Philosophical Perspectives on Infinity.</span></a><a class="discreet" title="View other works by Graham Robert Oppy" href="/web/20230531191753/https://philpapers.org/s/Graham Robert%20Oppy"><span class="name">Graham Robert Oppy</span></a> - <span class="pubYear">2006</span> - <span class="pubInfo"> New York: Cambridge University Press.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">This book is an exploration of philosophical questions about infinity. Graham Oppy examines how the infinite lurks everywhere, both in science and in our ordinary thoughts about the world. He also analyses the many puzzles and paradoxes that follow in the train of the infinite. Even simple notions, such as counting, adding and maximising present serious difficulties. Other topics examined include the nature of space and time, infinities in physical science, infinities in theories of probability and decision, the nature of<span id="OPPPPO-2-absexp"> (<span class="ll" onclick="$("OPPPPO-2-abstract2").show();$("OPPPPO-2-absexp").hide()">...</span>)</span><span id="OPPPPO-2-abstract2" style="display:none"> part/whole relations, mathematical theories of the infinite, and infinite regression and principles of sufficient reason. (<span class="ll" onclick="$("OPPPPO-2-abstract2").hide();$("OPPPPO-2-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-OPPPPO-2"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/general-relativity" rel="section">General Relativity</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-physical-science" rel="section">Philosophy of Physical Science</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/large-cardinals" rel="section">Large Cardinals</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/metaphysics-misc" rel="section">Metaphysics, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/metaphysics" rel="section">Metaphysics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics-misc" rel="section">Philosophy of Mathematics, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability-misc" rel="section">Philosophy of Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-infinite" rel="section">The Infinite</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/two-envelope-paradox" rel="section">Two-Envelope Paradox</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-action" rel="section">Philosophy of Action</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','OPPPPO-2')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=OPPPPO-2&proxyId=&u=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DFPU9tzW-2HAC%26printsec%3Dfront_cover"><i class="fa fa-download"></i> Direct download</a>   <div id="la-OPPPPO-2" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('OPPPPO-2')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-OPPPPO-2" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('OPPPPO-2','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/OPPPPO-2"><i class="fa fa-share-alt"></i> 35 citations</a>   <span class="eMsg" id="msg-OPPPPO-2"></span></div></div></li> <li id="eMARTAI-4" onclick="ee('click','MARTAI-4')" onmouseover="ee('over','MARTAI-4')" onmouseout="ee('out','MARTAI-4')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/MARTAI-4"><span class="articleTitle recTitle">Theories and inter-theory relations in Bošković.</span></a><a class="discreet" title="View other works by Ivica Martinovi" href="/web/20230531191753/https://philpapers.org/s/Ivica%20Martinovi"><span class="name">Ivica Martinovi</span></a> - <span class="pubYear">1990</span> - <span class="pubInfo"> <i class="pubName">International Studies in the Philosophy of Science</i> 4 (3):247 – 262.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">Abstract During 1745-1755 Bošković explicitly used the concept of scientific theory in three cases: the theory of forces existing in nature, the theory of transformations of geometric loci, and the theory of infinitesimals. The theory first mentioned became the famous theory of natural philosophy in 1758, the second was published in the third volume of his mathematical textbook Elementorum Universae Matheseos (1754), and the third theory was never completed, though Bošković repeatedly announced it from 1741 on. The treatment of continuity<span id="MARTAI-4-absexp"> (<span class="ll" onclick="$("MARTAI-4-abstract2").show();$("MARTAI-4-absexp").hide()">...</span>)</span><span id="MARTAI-4-abstract2" style="display:none"> and infinity in natural philosophy, geometry and infinitesimal analysis brought about inter?theory relations in Bošković's work during his Roman period. The two constructed theories of Bošković, the theory of forces and the theory of geometric transformations, directly influenced the idea for the construction of his third theory. These written theories refer to understanding and effective application of continuity and infinity in natural philosophy and geometry, and this task, according to Bošković, requires methodological support from the theory of infinitesimals. (<span class="ll" onclick="$("MARTAI-4-abstract2").hide();$("MARTAI-4-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-MARTAI-4"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/history-of-mathematics" rel="section">History of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/history-of-science-misc" rel="section">History of Science, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/general-philosophy-of-science" rel="section">General Philosophy of Science</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/history-philosophy-of-mathematics" rel="section">History: Philosophy of Mathematics</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-mathematics" rel="section">Philosophy of Mathematics</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/the-nature-of-theories" rel="section">The Nature of Theories</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/general-philosophy-of-science" rel="section">General Philosophy of Science</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','MARTAI-4')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=MARTAI-4&proxyId=&u=https%3A%2F%2Fdx.doi.org%2F10.1080%2F02698599008573365"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/MARTAI-4">(3 more)</a>   <div id="la-MARTAI-4" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('MARTAI-4')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-MARTAI-4" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('MARTAI-4','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/MARTAI-4"><i class="fa fa-share-alt"></i> 1 citation</a>   <span class="eMsg" id="msg-MARTAI-4"></span></div></div></li> <li id="eZAMOTI" onclick="ee('click','ZAMOTI')" onmouseover="ee('over','ZAMOTI')" onmouseout="ee('out','ZAMOTI')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/ZAMOTI"><span class="articleTitle recTitle">On the impossibility of events of zero probability.</span></a><a class="discreet" title="View other works by Asad Zaman" href="/web/20230531191753/https://philpapers.org/s/Asad%20Zaman"><span class="name">Asad Zaman</span></a> - <span class="pubYear">1987</span> - <span class="pubInfo"> <i class="pubName">Theory and Decision</i> 23 (2):157-159.</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="catsCon" id="ecats-con-ZAMOTI"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/axioms-of-probability" rel="section">Axioms of Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','ZAMOTI')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=ZAMOTI&proxyId=&u=https%3A%2F%2Fmpra.ub.uni-muenchen.de%2F9966%2F1%2FMPRA_paper_9966.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/ZAMOTI">(6 more)</a>   <div id="la-ZAMOTI" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('ZAMOTI')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-ZAMOTI" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('ZAMOTI','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/ZAMOTI"><i class="fa fa-share-alt"></i> 1 citation</a>   <span class="eMsg" id="msg-ZAMOTI"></span></div></div></li> <li id="eMACELD" onclick="ee('click','MACELD')" onmouseover="ee('over','MACELD')" onmouseout="ee('out','MACELD')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/MACELD"><span class="articleTitle recTitle">Expected Loss Divisibility Theorem.</span></a><a class="discreet" title="View other works by Rupert Macey-Dare" href="/web/20230531191753/https://philpapers.org/s/Rupert%20Macey-Dare"><span class="name">Rupert Macey-Dare</span></a> - <span class="pubYear">manuscript</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">This paper proposes and analyses the following theorem: For every total actual loss caused to a claimant with given probabilities by a single unidentified member of a defined group, there is a corresponding total expected loss, divisible and separable into discrete component expected sub-losses, each individually "caused" by a corresponding specific member of that defined group. Moreover, for every total estimated loss caused to a claimant in the past or present or prospectively in the future with estimable probabilities by one<span id="MACELD-absexp"> (<span class="ll" onclick="$("MACELD-abstract2").show();$("MACELD-absexp").hide()">...</span>)</span><span id="MACELD-abstract2" style="display:none"> or more unidentified members or causal agents from a defined group, the same result holds. The theorem is applied to Mesothelioma compensation claims. It provides a new justification for the proportional apportionment rule in Barker v Corus 2006 and an explanation of some paradoxical consequences of the rule and of the rule's overall fairness. (<span class="ll" onclick="$("MACELD-abstract2").hide();$("MACELD-absexp").show();">shrink</span>)</span></div><div class="catsCon" id="ecats-con-MACELD"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-law-misc" rel="section">Philosophy of Law, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-law" rel="section">Philosophy of Law</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability-misc" rel="section">Philosophy of Probability, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','MACELD')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=MACELD&proxyId=&u=http%3A%2F%2Fpapers.ssrn.com%2Fsol3%2Fpapers.cfm%3Fabstract_id%3D996992"><i class="fa fa-download"></i> Direct download</a>   <div id="la-MACELD" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('MACELD')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-MACELD" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('MACELD','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <span class="eMsg" id="msg-MACELD"></span></div></div></li> <li id="eBURAOO" onclick="ee('click','BURAOO')" onmouseover="ee('over','BURAOO')" onmouseout="ee('out','BURAOO')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/BURAOO"><span class="articleTitle recTitle">An outcome of the de finetti infinite lottery is not finite.</span></a><a class="discreet" title="View other works by Marc Burock" href="/web/20230531191753/https://philpapers.org/s/Marc%20Burock"><span class="name">Marc Burock</span></a> - <span class="pubYear">unknown</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">A randomly selected number from the infinite set of positive integers—the so-called de Finetti lottery—will not be a finite number. I argue that it is still possible to conceive of an infinite lottery, but that an individual lottery outcome is knowledge about set-membership and not element identification. Unexpectedly, it appears that a uniform distribution over a countably infinite set has much in common with a continuous probability density over an uncountably infinite set. </div><div class="catsCon" id="ecats-con-BURAOO"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/indifference-principles" rel="section">Indifference Principles</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-puzzles-misc" rel="section">Probabilistic Puzzles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','BURAOO')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=BURAOO&proxyId=&u=http%3A%2F%2Fphilsci-archive.pitt.edu%2Farchive%2F2605%2F1%2Finfinite_lottery.pdf"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/BURAOO">(3 more)</a>   <div id="la-BURAOO" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('BURAOO')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-BURAOO" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('BURAOO','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <span class="eMsg" id="msg-BURAOO"></span></div></div></li> <li id="eGWIPHA" onclick="ee('click','GWIPHA')" onmouseover="ee('over','GWIPHA')" onmouseout="ee('out','GWIPHA')" class="entry"><span class="citation"><a href="/web/20230531191753/https://philpapers.org/rec/GWIPHA"><span class="articleTitle recTitle">Probability, Hyperreals, Asymptotic Density, and God’s Lottery.</span></a><a class="discreet" title="View other works by Jeremy Gwiazda" href="/web/20230531191753/https://philpapers.org/s/Jeremy%20Gwiazda"><span class="name">Jeremy Gwiazda</span></a> - <span class="pubYear">unknown</span></span><span class="toggle" style="display:none" data-target="extras">details</span><div class="extras"><div class="abstract">Consider a subset, S, of the positive integers. What is the probability of selecting a number in S, assuming that each positive integer has an equal chance of selection? The purpose of this short paper is to provide an answer to this question. I also suggest that the answer allows us to determine the relative sizes of two subsets of the positive integers. </div><div class="catsCon" id="ecats-con-GWIPHA"><div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/decision-theoretic-puzzles" rel="section">Decision-Theoretic Puzzles</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-action" rel="section">Philosophy of Action</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinite-decision-theory" rel="section">Infinite Decision Theory</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-action" rel="section">Philosophy of Action</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/infinitesimals-and-probability" rel="section">Infinitesimals and Probability</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> <div><a class="catName" href="/web/20230531191753/https://philpapers.org/browse/probabilistic-puzzles-misc" rel="section">Probabilistic Puzzles, Misc</a><span class="catIn"> in </span><a class="catArea" href="/web/20230531191753/https://philpapers.org/browse/philosophy-of-probability" rel="section">Philosophy of Probability</a></div> </div><div class="options"><span title="Remove from this list" class="ll" onclick="removeFromList('5874','GWIPHA')"><i class="fa fa-times"></i> Remove from this list</span>   <a rel="nofollow" class="outLink" href="https://web.archive.org/web/20230531191753/https://philpapers.org/go.pl?id=GWIPHA&proxyId=&u=http%3A%2F%2Fphilsci-archive.pitt.edu%2Farchive%2F00005527%2F"><i class="fa fa-download"></i> Direct download</a> <a href="/web/20230531191753/https://philpapers.org/rec/GWIPHA">(5 more)</a>   <div id="la-GWIPHA" title="Export to another format" class="yui-skin-sam ldiv"> </div><span class="ll" onclick="showExports('GWIPHA')"><i class="fa fa-external-link"></i> Export citation<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>   <div id="ml-GWIPHA" class="yui-skin-sam ldiv"> </div><span title="Bookmark this publication" class="ll" onclick="showLists('GWIPHA','5874')"><i class="fa fa-bookmark"></i> Bookmark<img src="/web/20230531191753im_/https://philpapers.org/philpapers/raw/subind.gif"></span>  <a href="/web/20230531191753/https://philpapers.org/citations/GWIPHA"><i class="fa fa-share-alt"></i> 1 citation</a>   <span class="eMsg" id="msg-GWIPHA"></span></div></div></li> </ol> </div> </div> </div> </div> <div class="col-md-3 nopadding-xs nopadding-sm nopadding-right"> <div class="panel panel-default hidden-sm hidden-xs"> <div class="panel-body"> <div class="sideBox"> <div class="sideBoxH">Search inside</div> <div class="sideBoxC"> <form id="inside"> <input placeholder="Search keywords" size="19" class="searchInside" type="text" name="catq" value=""> <input type="hidden" name="sort" value="relevance"> <input type="hidden" name="uncat" value=""> <input type="hidden" name="setAside" value=""> <input type="submit" name="Search" value="Search"> </form> </div> </div> <div class="sideBox"> <div class="sideBoxH">Import / Add </div> <div class="sideBoxC"> <br> <script type="text/javascript">var currentList=5874;</script> <div class="paperadd" style="width:300px"> <form action="" onsubmit="alert('You need to select a paper from the results that show under the box.\nSearch using a surname followed by keywords.');return false"> Add an entry to this list: <input class="" id="authork1685544349x14756x108" name="authork" size="20" type="text" onfocus="if(this.value == 'Surname keyword') { this.value='' }" value="Surname keyword"> <input id="add-id" name="add-id" type="hidden"> </form> <div class="yui-skin-sam" id="auc-con1685544349x14756x108"></div> <script type="text/javascript"> watchForSymbol( { symbol:"xpa_yui_loaded", onSuccess: function() { var width = getWidth("400"); 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