Hume: Science, Logic, and Mathematics

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)

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.

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)

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)

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)

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)

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)

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)

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.

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 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 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)

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)

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)

„My sole purpose in this paper is to try and correct what I take to be a common misinterpretation of Hume’s opinions on mathematics. I shall not enquire whether he was right or wrong in holding these opinions. Nor shall I offer opinions of my own.“.

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view (...) of geometry is the distinction he draws between two standards of equality in extension. (shrink)

In the Treatise, David Hume denies the thesis that extension is infinitely divisible, even though it can be derived as a theorem of Euclidean geometry. This clearly shows that he rejects some of the theorems of Euclidean geometry. What is less clear is the extent to which he thinks geometry needs to be revised. It has been argued that Hume's rejection of infinite divisibility entails that most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection (...) theorem, are false, a view that is normally associated with Berkeley's earlier writings.I argue that Hume's denial of infinite divisibility is not incompatible with the Pythagorean theorem and other central theorems of. (shrink)

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science (...) is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. (shrink)

Hume seems to argue unconvincingly against the infinite divisibility of finite regions of space. I show that his conclusion is entailed by respectable metaphysical principles which he held. One set of principles entails that there are partless (unextended) things. Another set entails that these cannot be ordered so that an infinite number of them compose a finite interval.

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)

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)

Neo-logicism uses definitions and Hume's Principle to derive arithmetic in second-order logic. This paper investigates how much arithmetic can be derived using definitions alone, without any additional principle such as Hume's.

In the Foundations of Arithmetic, Frege famously developed a theory which today goes by the name of logicism - that it is possible to prove the truths of arithmetic using only logical principles and definitions. Logicism fell out of favor for various reasons, most spectacular of which was that the system, which Frege thought would definitively prove his thesis, turned out to be inconsistent. In the early 1980s a movement called neo-logicism was begun by Crispin Wright. Neo-logicism holds that Frege (...) was almost right, in that arithmetic can be proven in second-order logic using only definitions and one quasi-logical proposition, called Hume's Principle, which says that the number of Ps equals the number of Qs if and only if they can be put into one-to-one correspondence. There has been some controversy about the status of Hume’s Principle - for instance, whether it counts as a logical or analytic proposition. (See e.g. the similarly titled, “Is Hume’s Principle Analytic?, by Crispin Wright and George Boolos.) In this paper a different tack will be tried. Indeed Frege is almost right. He is almost right because a large part of arithmetic and number theory, or at the least a large part of something which looks like them, can indeed be generated using only logical principles and definitions, without the assumption of any quasi-logical assertion and in particular without Hume’s Principle. Specifically, logic will be taken as second-order logic with full comprehension and the addition of one distinguished 2-ary predicate “!”. A large amount of arithmetic and number theory will then be developed, using only (second-order) logical principles and definitions. It can thus be seen that the epistemological status of this large part of arithmetic is independent of the question of the status of Hume’s Principle. (shrink)

A number of formal constraints on acceptable abstraction principles have been proposed, including conservativeness and irenicity. Hume’s Principle, of course, satisfies these constraints. Here, variants of Hume’s Principle that allow us to count concepts instead of objects are examined. It is argued that, prima facie, these principles ought to be no more problematic than HP itself. But, as is shown here, these principles only enjoy the formal properties that have been suggested as indicative of acceptability if certain constraints on the (...) size of the continuum hold. As a result, whether or not these higher-order versions of Hume’s Principle are acceptable seems to be independent of standard (ZFC) set theory. This places the abstractionist in an uncomfortable dilemma: Either there is some inherent difference between counting objects and counting concepts, or new criteria for acceptability will need to be found. It is argued that neither horn looks promising. (shrink)

Hume’s discussion of space, time, and mathematics at T 1.2 appeared to many earlier commentators as one of the weakest parts of his philosophy. From the point of view of pure mathematics, for example, Hume’s assumptions about the infinite may appear as crude misunderstandings of the continuum and infinite divisibility. I shall argue, on the contrary, that Hume’s views on this topic are deeply connected with his radically empiricist reliance on phenomenologically given sensory images. He insightfully shows that, working within (...) this epistemological model, we cannot attain complete certainty about the continuum but only at most about discrete quantity. Geometry, in contrast to arithmetic, cannot be a fully exact science. A number of more recent commentators have offered sympathetic interpretations of Hume’s discussion aiming to correct the older tendency to dismiss this part of the Treatise as weak and confused. Most of these commentators interpret Hume as anticipating the contemporary idea of a finite or discrete geometry. They view Hume’s conception that space is composed of simple indivisible minima as a forerunner of the conception that space is a discretely (rather than continuously) ordered set. This approach, in my view, is helpful as far as it goes, but there are several important features of Hume’s discussion that are not sufficiently appreciated. I go beyond these recent commentators by emphasizing three of Hume’s most original contributions. First, Hume’s epistemological model invokes the “confounding” of indivisible minima to explain the appearance of spatial continuity. Second, Hume’s sharp contrast between the perfect exactitude of arithmetic and the irremediable inexactitude of geometry reverses the more familiar conception of the early modern tradition in pure mathematics, according to which geometry (the science of continuous quantity) has its own standard of equality that is independent from and more exact than any corresponding standard supplied by algebra and arithmetic (the sciences of discrete quantity). Third, Hume has a developed explanation of how geometry (traditional Euclidean geometry) is nonetheless possible as an axiomatic demonstrative science possessing considerably more exactitude and certainty that the “loose judgements” of the vulgar. (shrink)

In this essay, I argue that Hume?s theory of Quantitative and Numerical Philosophical Relations can be interpreted in a way which allows mathematical knowledge to be about a body of objective and necessary truths, while preserving Hume?s nominalism and the basic principles of his theory of ideas. Attempts are made to clear up a number of obscure points about Hume?s claims concerning the abstract sciences of Arithmetic and Algebra by means of re-examining what he says and what he could comfortably (...) have said about relations of qualitative resemblance. (shrink)

Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...) who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)

Don Garrett, Cognition and Commitment in Hume's Philosophy, New York and Oxford, Oxford University Press, 1997, pp. xiv + 270, Hb 40.00 ISBN 0-19-509721-1.

David Hume's central distinction in the Treatise and Enquiry is between relations of ideas and matters of fact. Although most of the attention in the secondary literature is on the latter, I am directly concerned with analyzing Hume's characterization of relations of ideas. In particular, I analyze Hume's attempt to secure certainty and necessity within his empiricist system since these are the defining characteristics of Hume's perfect species of knowledge--arithmetic. The focus, then, is on how Hume justifies the special status (...) of arithmetical knowledge. ;Two interpretations of Hume's philosophy of arithmetic have been offered in the secondary literature. But I argue that both are inadequate since, among other things, they fail to render arithmetic consistent with Hume's general theory of meaning nor do they connect the certainty of arithmetic with Hume's discussion of 1-1 correspondence. After critically analyzing these two positions, I argue for a new interpretation. I show that Hume's aversion to abstraction and infinite divisibility leads to a constructivist account of number--one which bears a strong resemblance to mathematical intuitionism. ;Not surprisingly, certainty and necessity are difficult to obtain in the Humean system. The subjective nature of psychological links of association precludes certainty. So Hume needs to connect arithmetic with something outside of his phenomenological experience. And I argue that this something else is the metaphysical makeup of space. In Book I, Part II of the Treatise, Hume adapts Zeno's metrical paradox to demonstrate that objective reality must consist of absolute simples, which Hume calls "units". Hume then uses these units as the building blocks for his constructivist theory of number. For only after establishing the existence of units, does Hume make his central distinction between relations of ideas and matters of fact and deploy his constructivism. ;Although problematic, Hume's philosophy of arithmetic provides us with a new perspective on his empiricism. Hume's frequently ignored discussion of space is found to impact directly his views on arithmetic. And his commitment to absolute simples provides direct evidence that Hume is a skeptical realist, having more in common with Locke than previously recognized. (shrink)