Hume: Science, Logic, and Mathematics

This article centers on Hume’s position on the intelligibility of natural philosophy. To that end, the controversy surrounding universal gravitation shall be scrutinized. It is very well-known that Hume sides with the Newtonian experimentalist approach rather than with the Leibnizian demand for intelligibility. However, what is not clear is Hume’s overall position on the intelligibility of natural philosophy. It shall be argued that Hume declines Leibniz’s principle of intelligibility. However, Hume does not eschew intelligibility altogether; his concept of causation itself (...) stipulates mechanical intelligibility. (shrink)

This book contextualizes David Hume's philosophy of physical science, exploring both Hume's background in the history of early modern natural philosophy and its subsequent impact on the scientific tradition.

The last few decades have witnessed intense debates in Hume scholarship concerning Hume’s account of causation. At the core of the “old–new Hume” debate is the question of whether causation for Hume is more than mere regularity, in particular, whether Hume countenances necessary connections in mind-independent nature. This chapter assesses this debate against the background of Hume’s “foundational project” in the Treatise. The question of the role and import of Hume’s account of the idea of cause is examined and compared (...) with Hume’s treatment of other ideas. (shrink)

Given the sharp distinction that follows from Hume’s Fork, the proper epistemic status of propositions of mixed mathematics seems to be a mystery. On the one hand, mathematical propositions concern the relation of ideas. They are intuitive and demonstratively certain. On the other hand, propositions of mixed mathematics, such as in Hume’s own example, the law of conservation of momentum, are also matter of fact propositions. They concern causal relations between species of objects, and, in this sense, they are not (...) intuitive or demonstratively certain, but probable or provable. In this article, I argue that the epistemic status of propositions of mixed mathematics is that of matters of fact. I wish to show that their epistemic status is not a mystery. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle, unlike the propositions of pure mathematics. (shrink)

Though it is well known that Hume composed his Treatise and the first version of his essay On Miracles during his stay in La Flèche, this fact has not received the attention it surely deserves. What may have induced a gentleman from Edinburgh to bury himself in the library of a Jesuit college? While every historian of early modern philosophy is quick to credit La Flèche with having provided Descartes’s education, Burton’s amazement that Hume himself never alludes to this unique (...) circumstance is still suggestive. I shall argue that what attracted Hume’s attention were the scholastic items contained in every good Jesuit library. (shrink)

The subject of this essay-based dissertation is Hume’s natural philosophy. The dissertation consists of four separate essays and an introduction. These essays do not only treat Hume’s views on the topic of natural philosophy, but his views are placed into a broader context of history of philosophy and science, physics in particular. The introductory section outlines the historical context, shows how the individual essays are connected, expounds what kind of research methodology has been used, and encapsulates the research contributions of (...) the essays. The first essay treats Newton’s experimentalist methodology in gravity research and its relation to Hume’s causal philosophy. It is argued that Hume does not see the relation of cause and effect as being founded on a priori reasoning, similar to the way in which Newton criticized non-empirical hypotheses about the causal properties of gravity. Contrary to Hume’s rules of causation, the universal law does not include a reference either to contiguity or succession, but Hume accepts it in interpreting the force and the law of gravity instrumentally. The second article considers Newtonian and non-Newtonian elements in Hume more broadly. He is sympathetic to many prominently Newtonian themes in natural philosophy, such as experimentalism, critique of hypotheses, inductive proof, and the critique of Leibnizian principles of sufficient reason and intelligibility. However, Hume is not a Newtonian philosopher in many respects: his conceptions regarding space and time, the vacuum, the specifics of causation, the status of mechanism, and the reality of forces differ markedly from Newton’s related conceptions. The third article focuses on Hume’s Fork and the proper epistemic status of propositions of mixed mathematics. It is shown that the epistemic status of propositions of mixed mathematics, such as those concerning laws of nature, is that of matters of fact. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle. The fourth article analyzes Einstein’s acknowledgement of Hume regarding special relativity. The views of the scientist and the philosopher are juxtaposed, and it is argued that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts and a relationist ontology regarding space and time. (shrink)

Einstein acknowledged that his reading of Hume influenced the development of his special theory of relativity. In this article, I juxtapose Hume’s philosophy with Einstein’s philosophical analysis related to his special relativity. I argue that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts, and a relationist ontology regarding space and time. The main thesis of this article is that these two points are intertwined in Hume and Einstein.

Hume's Principle, dear to neo-Logicists, maintains that equinumerosity is both necessary and sufficient for sameness of cardinal number. All the same, Whitehead demonstrated in Principia Mathematica's logic of relations that Cantor's power-class theorem entails that Hume's Principle admits of exceptions. Of course, Hume's Principle concerns cardinals and in Principia's ‘no-classes’ theory cardinals are not objects in Frege's sense. But this paper shows that the result applies as well to the theory of cardinal numbers as objects set out in Frege's Grundgesetze. (...) Though Frege did not realize it, Cantor's power-theorem entails that Frege's cardinals as objects do not always obey Hume's Principle. (shrink)

The paper formulates and proves a strengthening of 'Frege's Theorem', which states that axioms for second-order arithmetic are derivable in second-order logic from Hume's Principle, which itself says that the number of Fs is the same as the number of Gs just in case the Fs and Gs are equinumerous. The improvement consists in restricting this claim to finite concepts, so that nothing is claimed about the circumstances under which infinite concepts have the same number. 'Finite Hume's Principle' also suffices (...) for the derivation of axioms for arithmetic and, indeed, is equivalent to a version of them, in the presence of Frege's definitions of the primitive expressions of the language of arithmetic. The philosophical significance of this result is also discussed. (shrink)

In this note we derive Robinson???s Arithmetic from Hume???s Principle in the context of very weak theories of classes and relations.

One recent `neologicist' claim is that what has come to be known as "Frege's Theorem"–the result that Hume's Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate–reinstates Frege's contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume's Principle itself. The present paper reviews five misgivings that developed in various of George Boolos's writings. It observes that each of them really concerns not `analyticity' but either the truth of Hume's Principle or our entitlement (...) to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos's objections–the problem of Bad Company–and outlines a proof of the principle $N^q$, an important part of the defense of the claim that what follows from Hume's Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself. (shrink)

In this paper, the author discusses the feasibility of constructing a Humean model of the psychological realities of categorical propositions and syllogistic deduction by employing only Hume’s kinds of “ideas” and kinds of mental operations on ideas which Hume explicitly or implicitly postulated in his theory of discursive thinking.

In this paper, I shall discuss several topics related to Frege's paradigms of second-order abstraction principles and his logicism. The discussion includes a critical examination of some controversial views put forward mainly by Robin Jeshion, Tyler Burge, Crispin Wright, Richard Heck and John MacFarlane. In the introductory section, I try to shed light on the connection between logical abstraction and logical objects. The second section contains a critical appraisal of Frege's notion of evidence and its interpretation by Jeshion, the introduction (...) of the course-of-values operator and Frege's attitude towards Axiom V, in the expression of which this operator occurs as the key primitive term. Axiom V says that the course-of-values of the function f is identical with the course-of-values of the function g if and only if f and g are coextensional. In the third section, I intend to show that in Die Grundlagen der Arithmetik Frege hardly could have construed Hume's Principle as a primitive truth of logic and used it as an axiom governing the cardinality operator as a primitive sign. HP expresses that the number of Fs is identical with the number of Gs if and only if F and G are equinumerous. In the fourth section, I argue that Wright falls short of making a convincing case for the alleged analyticity of HP. In the final section, I canvass Heck's arguments for his contention that Frege knew he could deduce the simplest laws of arithmetic from HP without invoking Axiom V. I argue that they do not carry conviction. I conclude this section by rejecting an interpretation concerning HP suggested by MacFarlane. (shrink)

In a recent article, I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided (...) by Cantor’s cardinality assignments. In this talk, I generalize some specific worries emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The talk has four main parts. The first takes a historical look at 19th-century attributions of equality of numbers in terms of one-one correlations and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions. (shrink)

This is a commentary on Adrian Heathcote’s interesting paper ‘Hume’s Master Argument’. Heathcote contends that No-Ought-From-Is is primarily a logical thesis, a ban on Is/Ought inferences which Hume derives from the logic of Ockham. NOFI is thus a variation on what Heathcote calls ‘Hume’s Master Argument’, which he also deploys to prove that conclusions about the future (and therefore a-temporal generalizations) cannot be derived by reason from premises about the past, and that conclusions about external objects or other minds cannot (...) be derived by reason from premises about impressions. Heathcote raises an important question. Having (apparently) argued that our inductive inferences are not justified by reason, Hume puts them down to Custom, and seems to suggest that we OUGHT to indulge this propensity but NOt the superstitious propensities that lead to religious belief. (Query: Why is it right to indulge one non-rational propensity but not the others?) Finally Heathcote argues that just as there are valid, but not formally valid, arguments taking us from claims about inferential relations to claims about what we ought to believe, so there may be valid, but not formally valid, arguments taking us from factual claims about some situation to claims about what we ought to do. -/- I reply that Hume does indeed have a Master Argument and that it does rely on logical principles but not on the logic of Ockham which had been largely forgotten by Hume’s day. Instead Hume relies on the idea widely believed in the 18th Century and taught to Hume at Edinburgh by his logic Professor Colin Drummond, that in a logically valid argument the conclusion is contained in the premises. I reconstruct Hume’s Master Argument using this principle. I draw a careful distinction between two theses: 1) that we cannot get from non-moral premises to moral conclusions with the aid of logic alone and 2) that we cannot get from non-moral premises to moral conclusions with aid of analytic bridge principles. Hume believed the first but not the second. What then is the role of NOFI in the larger argument of the Treatise? To show that the truths of ethics cannot be derived via logic from self-evident truths of some other kind and thus that they are not demonstrable. How can we make sense of Hume’s apparent belief that it is sometimes right to transcend reason and sometimes not? In the case of Custom, we live in a world governed by causal regularities, and, in such a world, induction is in fact a fairly reliable belief-forming mechanism. Thus a suitably qualified spectator (one aware of the kind of world we live in) would tend to approve of indulging it, even if it cannot be justified by reason. However, our superstitious propensities are (and can be known to be) unreliable, since they produce different and inconsistent results in different people. Thus it is it is wrong (something a suitably qualified spectator would disapprove of) to indulge the faculty of Superstition. I also take issue with Heathcote’s penchant for valid, but not formally valid, inferences. I supply the missing premises for Heathcote’s Is/Ought inferences and argue that they are either not true or not necessary. (shrink)

To support my Phd theses and results of my grant research in 1999, I asked 1) prominent chemist Antonín Holý, author of substances to treat hepatitis and HIV, about the indivisibility of the art and science (published in Slovak Narodna Obroda and Czech blisty,cz), 2) the distinguished economist William Baumol about the alternative activities (published in Slovak Nove Slovo, Czech Respekt and blisty.cz), 3) Nobel Laureate Clive Granger about the significance of the economics (published in 2004 in Czech weekly Tyden). (...) The interviews were exhibited in Holland Park, W8 6LU, The Ice House between 18. Oct - 3. Nov. 2013. (shrink)

Hume’s views concerning the existence of body or external objects are notoriously difficult and intractable. The paper sheds light on the concept of body in Hume’s Treatise by defending three theses. First, that Hume’s fundamental tenet that the only objects that are present to the mind are perceptions must be understood as methodological, rather than metaphysical or epistemological. Second, that Hume considers legitimate the fundamental assumption of natural philosophy that through experience and observation we know body. Third, that many of (...) the contradictions and difficulties that interpreters attribute to Hume’s concept of body should be attributed instead, as Hume does, to every system of philosophy. (shrink)

The paper addresses two difficulties that arise in Treatise 1.2.5. First, Hume appears to be inconsistent when he denies that we have an idea of a vacuum or empty space yet allows for the idea of an “invisible and intangible distance.” My solution to this difficulty is to develop the overlooked possibility that Hume does not take the invisible and intangible distance to be a distance at all. Second, although Hume denies that we have an idea of a vacuum, some (...) texts in Treatise 1.2.5 are taken by interpreters to suggest that Hume nonetheless believes that there are vacuums in nature. I discuss the relevant texts and defend the view that Hume does not in fact countenance belief in vacuums. I conclude by outlining an interpretation of Hume’s intention in the Treatise that allows us to understand his discussion of ideas as having implications for the sciences. (shrink)

Hume appeals to different kinds of certainties and necessities in the Treatise. He contrasts the certainty that arises from intuition and demonstrative reasoning with the certainty that arises from causal reasoning. He denies that the causal maxim is absolutely or metaphysically necessary, but he nonetheless takes the causal maxim and ‘proofs’ to be necessary. The focus of this paper is the certainty and necessity involved in Hume’s concept of knowledge. I defend the view that intuitive certainty, in particular, is certainty (...) of the invariability or necessity of relations between ideas. Against David Owen and Helen Beebee, I argue that the certainty involved in intuition depends on the activity of the mind. I argue, further, that understanding this activity helps us understand more clearly one of Hume’s most important theses, namely that experience is the source of a distinct kind of certainty and of necessity. (shrink)

In the Introduction to the Treatise Hume very enthusiastically announces his project to provide a secure and solid foundation for the sciences by grounding them on his science of man. And Hume indicates in the Abstract that he carries out this project in the Treatise. But most interpreters do not believe that Hume's project comes to fruition. In this paper, I offer a general reading of what I call Hume's ‘foundational project’ in the Treatise, but I focus especially on Book (...) 1. I argue that in Book 1 much of Hume's logic is put in the service of the other sciences, in particular, mathematics and natural philosophy. I concentrate on Hume's negative thesis that many of the ideas central to the sciences are ideas that we cannot form. For Hume, this negative thesis has implications for the sciences, as many of the texts I discuss make evident. I consider and criticize different proposals for understanding these implications: the Criterion of Meaning and the ‘Inconceivability Principle’. I introduce what I call Hume's ‘No Reason to Believe’ Principle, which I argue captures more adequately the link Hume envisions between his logic, in particular his examination of ideas, and the other sciences. (shrink)

In this paper, the author discusses the feasibility of constructing a Humean model of the psychological realities of categorical propositions and syllogistic deduction by employing only Hume’s kinds of “ideas” and kinds of mental operations on ideas which Hume explicitly or implicitly postulated in his theory of discursive thinking.

We call Frege's discovery that, in the context of second-order logic, Hume's principle-viz., The number of Fs = the number of Gs if, and only if, F a G, where F a G (the Fs and the Gs are in one-to-one correspondence) has its usual, second-order, explicit definition-implies the infinity of the natural numbers, Frege's theorem. We discuss whether this theorem can be marshalled in support of a possibly revised formulation of Frege's logicism.

Hume argues that the idea of duration is just the idea of the manner in which several things in succession are arrayed. In other words, the idea of duration is the idea of successiveness. He concludes that all and only successions have duration. Hume also argues that there is such a thing as a steadfast object—something which co-exists with many things in succession, but which is not itself a succession. Thus, it seems that Hume has committed himself to a contradiction: (...) A steadfast object lacks duration because it is not a succession, but has duration because it co-exists with something which has duration. I am not going to discuss why Hume thinks these things. My goal is simply to show that what he thinks is consistent. To do so, I will offer a Humean temporal logic. (shrink)

We can modify Hume’s Principle in the same manner that George Boolos suggested for modifying Frege’s Basic Law V. This leads to the principle Small Hume. Then, we can show that Small Hume is interderivable with Hume’s Principle.

This paper presents a new account of Hume’s “probability of causes”. There are two main results attained in this investigation. The first, and perhaps the most significant, is that Hume developed – albeit informally – an essentially sound system of probabilistic inductive logic that turns out to be a powerful forerunner of Carnap’s systems. The Humean set of principles include, along with rules that turn out to be new for us, well known Carnapian principles, such as the axioms of semiregularity, (...) symmetry with respect to individuals (exchangeability), predictive irrelevance and positive instantial relevance. The second result is that Hume developed an original conception of probability, which is subjective in character, although it differs from contemporary personalistic views because it includes constraints that are additional to simple consistency and do not vary between different persons. The final section is a response to Gower’s thesis, by which Hume’s probability of causes is essentially non-Bayesian in character. It is argued that, on closer examination, Gower’s reading of the relevant passages is untenable and that, on the contrary, they are in accordance with the Bayesian reconstruction presented in this paper. (shrink)

In the Treatise of Human Nature, David Hume mounts a spirited assault on the doctrine of the infinite divisibility of extension, and he defends in its place the contrary claim that extension is everywhere only finitely divisible. Despite this major departure from the more conventional conceptions of space embodied in traditional geometry, Hume does not endorse any radical reform of geometry. Instead Hume espouses a more conservative approach, claiming that geometry fails only “in this single point” – in its purported (...) proofs of infinite divisibility – while “all of its other arguments” remain intact. -/- In this paper, after laying out the prima facie case for Hume’s radical challenge to traditional geometry, I consider five strategies for blocking the arguments for infinite divisibility while conserving most of geometry. I show that each of these interpretive strategies suffers from serious substantive problems, and so none of them delivers an interpretation of Hume’s account that provides him with a way of blocking the geometric arguments for infinite divisibility while sustaining his geometric conservatism. (shrink)

Sections “Introduction: Hume’s Principle, Basic Law V and Cardinal Arithmetic” and “The Julius Caesar Problem in Grundlagen—A Brief Characterization” are peparatory. In Section “Analyticity”, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle, bearing in mind that with its analytic or non-analytic status the intended logical foundation of cardinal arithmetic stands or falls. Section “Thought Identity and Hume’s Principle” is concerned with the two criteria of thought identity that Frege states in 1906 and (...) their application to Hume’s Principle. In Section “The Nature ofion: A Critical Assessment of Grundlagen, §64”, I scrutinize Frege’s characterization of abstraction in Grundlagen, §64 and criticize in this context the currently widespread use of the terms “recarving” and “reconceptualization”. Section “Frege’s Proof of Hume’s Principle” is devoted to the formal details of Frege’s proof of Hume’s Principle. I begin by considering his proof sketch in Grundlagen and subsequently reconstruct in modern notation essential parts of the formal proof in Grundgesetze. In Section “Equinumerosity and Coextensiveness: Hume’s Principle and Basic Law V Again”, I discuss the criteria of identity embodied in Hume’s Principle and in Basic Law V, equinumerosity and coextensiveness. In Section “Julius Caesar and Cardinal Numbers—A Brief Comparison Between Grundlagen and Grundgesetze ”, I comment on the Julius Caesar problem arising from Hume’s Principle in Grundlagen and analyze the reasons for its absence in this form in Grundgesetze. I conclude with reflections on the introduction of the cardinals and the reals by abstraction in the context of Frege’s logicism. (shrink)

Debunking arguments against both moral and mathematical realism have been pressed, based on the claim that our moral and mathematical beliefs are insensitive to the moral/mathematical facts. In the mathematical case, I argue that the role of Hume’s Principle as a conceptual truth speaks against the debunkers’ claim that it is intelligible to imagine the facts about numbers being otherwise while our evolved responses remain the same. Analogously, I argue, the conceptual supervenience of the moral on the natural speaks presents (...) a difficulty for the debunker’s claim that, had the moral facts been otherwise, our evolved moral beliefs would have remained the same. (shrink)

In recent work Bruno Whittle has presented a new challenge to the Cantorian idea that there are different infinite cardinalities. Most challenges of this kind have tended to focus on the status of the axioms of standard set theory; Whittle’s is different in that he focuses on the connection between standard set theory and intuitive concepts related to cardinality. Specifically, Whittle argues we are not in a position to know a principle I call the Quantificational Hume Principle, which connects the (...) application of intuitive, quantificational cardinality concepts to claims involving the existence of functions. This paper responds to Whittle’s skeptical arguments by providing an argument justifying the QHP. (shrink)

According to Hume’s principle, a sentence of the form ⌜The number of Fs = the number of Gs⌝ is true if and only if the Fs are bijectively correlatable to the Gs. Neo-Fregeans maintain that this principle provides an implicit definition of the notion of cardinal number that vindicates a platonist construal of such numerical equations. Based on a clarification of the explanatory status of Hume’s principle, I will provide an argument in favour of a nominalist construal of numerical equations. (...) The neo-Fregean objections to such a construal will be examined and rejected. And the implications of the nominalist construal for the use of numerals and for the understanding of ontological questions for the existence of numbers will be spelled out. (shrink)

The question of the analyticity of Hume’s Principle is central to the neo-logicist project. We take on this question with respect to Frege’s definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege’s definition of number, it isn’t analytic, and if HP is taken to be (...) primitive there is only a very narrow range of circumstances where it might be taken to be analytic. The latter discussion also sheds some light on the connections between the Bad Company and Caesar objections. (shrink)