# Has philosophy ever clarified mathematics?

I've recently been reading some standard textbooks on the philosophy of mathematics, and I've become quite frustrated that (surely due to my own limitations) I don't seem to be gleaning any mathematical insights from them.

My naïve expectation would be that philosophy might take a difficult construction or proof, and clarify it by isolating the key ideas behind it. Having isolated the key ideas, philosophy might then highlight their relevance and thus point the way forward. Beyond this, I would hope that philosophy might elucidate the true meaning' of axioms and of definitions by examining their ontology in a wider context.

In reality, to the best of my knowledge (please prove me wrong!) both of the above tasks seem to be carried out exclusively by mathematicians, physicists, computer scientists, and other natural scientists, as far as I can see. To play the devil's advocate, philosophy seems to me like it might historically have largely played an opposite role, labeling certain objects as "unreal" and "unnatural" which in fact later turned out to be fruitful to study (negative numbers, irrational numbers, complex numbers...).

Question: Has it ever happened that philosophy has elucidated and clarified a mathematical concept, proof, or construction in a way useful to research mathematicians?

Philosophers have created much new mathematics (e.g. the work of C.S. Peirce, much of which is bona fide mathematical research), but the question is not about this, but rather about philosophy as practiced by philosophers providing elucidation, explanation, and clarification of existing mathematics.

• I have the impression that Lawvere thought a lot about philosophy of mathematics and that those thoughts strongly informed his work in category theory. I'm not really qualified to discuss this though. – Qiaochu Yuan Oct 1 '14 at 8:48
• My naive expectation is very different from yours: I would expect philosophers of mathematics to be interested in foundational questions about the scope and consistency of axiomatic thinking, the status of mathematical truth, the relationship between mathematical models and the phenomena that they claim to model, etc. In my experience philosophy doesn't try to extract the "key ideas" in any discipline - those are determined by experts within the discipline itself - rather, it analyzes the assumptions, methods, and practices that characterize the discipline. – Paul Siegel Oct 1 '14 at 16:02
• @PaulSiegel Your comment should be an answer... – An old man in the sea. Oct 1 '14 at 22:40
• Often the clarifying role of philosophy has been in a double negative sense: good philosophy can help mathematics by preventing bad philosophy from muddying the conceptual waters (e.g., to pick just one such confusion that is no longer an issue, the philosophical controversy about the ontological status of non-Euclidean geometry). – Terry Tao Oct 2 '14 at 13:56
• @PaulSiegel, As a philosopher, your answer strikes me as spot on. I wouldn't imagine a philosopher of math is going to try to help mathematicians do their mathematics better. Instead, I'd expect the philosopher of math to be interested in drawing big pictures about the nature of mathematical knowledge, or else in asking questions that mathematicians themselves don't care too much about like, "How is it possible that abstract entities like numbers enter into the explanation of physical phenomena?" – shane Oct 4 '14 at 13:38

Two points: one, firstly understanding mathematical processes can be of immense pedagogical value. See e.g. Polya's How to Solve It (and he wrote a more academic book with these themes), or Lakatos' Proofs and Refutations. I found this book New Directions in the Philosophy of Mathematics which had interesting essays as well.

Secondly, remember that broadly the point of philosophy is to make things not philosophy. In extremely simplistic historical terms, once natural philosophy becomes rigorous it becomes science, once philosophy of language became rigorous it became linguistics, and today we're seeing philosophy of mind turn to neuroscience.

So philosophy that elucidates mathematics is simply... mathematics. Most obviously Russell and the development of set theory. Modernly I don't know: I think the interesting stuff is happening at computer science/philosophy and physics/philosophy which trickles into mathematics. I'm posting this largely because I think the question is slightly broken because philosophy doesn't really work to clarify a field where it has already been clarified.

• I agree completely with this point. – Joel David Hamkins Oct 1 '14 at 7:53
• I like very much the phrase "the point of philosophy is to make things not philosophy" and the accompanying elucidation. Are you quoting somebody here? If so, whom? – Ian Morris Oct 1 '14 at 17:56
• @Ian Morris: Well, Bertrand Russell famously said, "As soon as definite knowledge concerning any subject becomes possible, this subject ceases to be called philosophy, and becomes a separate science." Similar sentiments have been voiced by many in the positivist tradition. In our day, scientists such as Steven Pinker and Lawrence Krauss have popularized similar views of philosophy. Of course, like everything in philosophy, this view of philosophy is controversial! – Timothy Chow Oct 2 '14 at 2:00
• Philosophy of mind didn't become neuroscience and philosophy of language didn't become linguistics either. Both of those departments of philosophy are empirically informed, of course. Nobody wants to propose a theory of mind or a theory about the relation between language and the world that isn't empirically adequate. But there are properly philosophical questions like What is meaning? or How can one physical state be about another? It is hard to see how future empirical discoveries could shed any light on these questions. – shane Oct 4 '14 at 13:34
• "the point of philosophy is to make things not philosophy" This is possiby the best defense of philosophy I've ever heard. – RBarryYoung Oct 6 '14 at 20:20

I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable. Even Gödel had despaired to ever have a formal notion of computability, because he had expected that for any such formal notion of computability, we would be able to diagonalize against it and thereby find a function that was computable in an intuitive sense, but not with respect to the formal notion. This was true of the class of primitive recursive functions and other extensions of that idea.

Meanwhile, Turing proceeded on a mainly philosophical level to consider what it was that a human did when undertaking a computation, imagining a person with paper and pencil and plenty of time, following a rote computational procedure, and was thereby led to his notion of Turing machine, which led to the fields of computability theory, complexity theory and so on.

• Thanks for this answer! In the same vein, perhaps one might mention Pearl's "Causality". Conversely, I could perhaps argue that Turing was a mathematician doing mathematics, perhaps with philosophical motivation, as opposed to a philosopher doing philosophy. Also that it was new mathematics. Would you agree with this? – Daniel Moskovich Oct 1 '14 at 6:42
• Well, I believe that Turing's derivation of the definition of Turing machine is usually regarded mainly as a piece of philosophy, rather than mathematics. For example, it is often read in philosophy seminars, but not usually in mathematical seminars. But I argue in my answer at mathoverflow.net/a/172848/1946 that the subjects do not admit a clear separation---they blend one into the other. Often, the mathematics I find most interesting has a philosophical aspect, and conversely. – Joel David Hamkins Oct 1 '14 at 6:47
• Nowadays, of course, the Turing machine definition itself, which was the outcome of the philosophical analysis, is taken purely mathematically. – Joel David Hamkins Oct 1 '14 at 6:49
• It seems to me that Turing created and investigated a mathematical model of computation. Many other mathematicians (and other scientists) created and investigated mathematical models of many other things. It is not clear to me why this one should count as philosphy while others do not (or maybe the others do too?). [I would not claim I read Turing's paper, but I browsed it.] – user9072 Oct 1 '14 at 12:32
• @user9072: Sophie Germain built a mathematical model of a physical phenomenon, the vibration of elastic surfaces, and is therefore regarded as a mathematical physicist. Alan Turing built a mathematical model of a philosophical phenomenon, effective calculation, and could therefore I think be regarded as a mathematical philosopher. As an aside, just looking at some of Turing's most famous ideas, it seems to me that Turing's work stretches seamlessly from mathematics to philosophy—from the Turing machine to the Church-Turing thesis to the Turing test. – Vectornaut Jun 17 '16 at 4:04

As already remarked by others: If one tries a narrow interpretation of your question, you are asking a lot. You want someone whose specialty is not mathematics to elucidate a mathematical argument in a way useful and interesting for mathematician in their practice of mathematics, without creating new mathematics.

Interpreted in a slightly broader sense, the first men to come to my mind are Bolzano and Frege.

Bolzano was certainly a philosopher (and priest). His work on what we now consider set theory can very well be regarded as a philosophical elucidation of the mathematical treatment of, I would say, infinite sets, only that prior to Bolzano noone has really thought in these terms! Of course, there is other work by Bolzano, which more purely qualifies as mathematics, as his efforts to put analysis on a firm foundation and his example of a continuous and nowhere differentiable function, which first of all required clear concepts of these terms! But it should be clear that these questions had then a strong philosophical bend; only now that we take these modes of thinking for granted, we can claim that they are just part of mathematics and not philosophy.

Frege is considered by many analytic philosophers as one of the foremost philosophers of the 19th century, although one has to remark that he studied mathematics. In his Begriffsschrift he essentially invented formal predicate logic. In his later work he applied logic to the foundations of mathematics. Formal logic has certainly done a lot for mathematics. Of course, you could claim that formal logic is really the creation of new mathematics, but I think it was not primarly so. Coming up with formal logic was foremost an act of thinking hard about what an argument is - a typical philosophical activity.

There also names like Quine, Kripke and Dana Scott, where part of their works blurs the distinction between set theory, logic, philosophy and mathematics.

Many of the mathematicians active in foundations had also a strong philosophical interest. I want just to mention the names Cantor, Hausdorff, Gödel and MacLane. Their philosophical interests had certainly influence on their mathematics, although this is probably hard to prove.

Even one step further: Mathematicians have certainly done interesting philosophical work on mathematics, even if they do not claim so. I just want to mention Tao's What is good mathematics? and Thurston's On proof and progress in mathematis.

Even more ordinary mathematicians use from time to time phrases like "from a philosophical point of view" in their mathematical musings. One might mark this just as a non-rigorous mode of mathematical thinking, but also as a philosophical-bend mode of thinking - both are correct at the same time, I suppose.

• +1 I think it's hard to overestimate the importance of the Begriffsschrift – Tom Harris Oct 8 '14 at 9:24

From this article in Wikipedia : "La Géométrie was published in 1637 as an appendix to Discours de la méthode (Discourse on Method), written by René Descartes. In the Discourse, he presents his method for obtaining clarity on any subject. La Géométrie and two other appendices also by Descartes, the Optics and the Meteorology, were published with the Discourse to give examples of the kinds of successes he had achieved following his method".

• Moreover, Descartes presented (and was to some extent motivated by) an elaborate argument about which curves can be 'known' which has philosophical aspects; see Bos's Redefining Geometrical Exactness. – Marius Kempe Mar 25 '15 at 17:55

In order to address this question, I think it is important to first take a step back and examine with a critical eye something that we normally take for granted, namely the professionalization and compartmentalization of academic departments. It is common to think of "philosophy" as "that which is practiced by professional philosophers" and "mathematics" as "that which is practiced by professional mathematicians." However, in my opinion, the divvying up of academic "turf" is driven more by sociological and economic factors than by any intrinsic divisions in the intellectual subject matter. To put it crassly (and somewhat exaggeratedly), my fellow academicians and I stand to extract more money and prestige from the rest of society if we agree to slice up the pie in a certain way and not fight too much internally over the division, conserving our energy to be directed outwards.

But if we actually want to understand the true relationship between "philosophy" and "mathematics," we shouldn't confuse ourselves by insisting that what mathematicians do is by definition "not philosophy" and that "philosophy" is by definition what is done by people whose paycheck comes from a philosophy department of an academic institution.

If we accept this point of view, then we should have no qualms about using the word "philosophy" to describe the sorts of activities that you described as being "carried out exclusively by mathematicians and scientists." In that sense, philosophy has always clarified mathematics and will continue to clarify mathematics.

Now, you might still have a lingering question, which might be rephrased as follows: "Does the kind of philosophy of mathematics that happens to currently be the bread and butter of those people who happen to be paid by academic philosophy departments stand to offer any clarification of mathematics?" Phrased this way, we can see that it is more of a sociological question (and a historical question, since the political agreements about who owns what turf change over time) rather than a question about philosophy and mathematics per se. But still, we might be interested in the answer.

Generally speaking, the kind of thing that is covered in "textbooks of philosophy of mathematics" interacts with mathematics via the foundations of mathematics in general. The influence that this kind of philosophy has had in clarifying mathematics in general is clearest if we look back at the early 20th century. For example, today we enjoy an immensely clearer notion of what a "proof" is than anyone had in the 19th century, and that is thanks to the clarifying work of those who worked in foundations. We also have a much clearer notion of the distinction between constructive and non-constructive mathematics, thanks to foundational work surrounding the axiom of choice, intuitionism, uncomputability, etc. For a modern example, Voevodsky and his collaborators are now pushing homotopy type theory as their preferred approach to foundations. (Admittedly most of their paychecks don't come from philosophy departments but until recently, I think most authors of books and papers on type theory were paid by philosophy departments.)

In my opinion there is a continuum between foundations of mathematics in general and foundations of specific areas of mathematics (e.g., the foundations of algebraic geometry—think of topoi), but if you insist on defining "philosophy" according to which pot of money the paycheck comes from, then the foundations of algebraic geometry won't count because our society is set up so that such people experience pressure to migrate to the mathematics department. Thus so-called "philosophy of mathematics" is more-or-less forced to limit itself to clarifying the foundations of mathematics in general.

• I don't know if it's too much to expect for philosophers to clarify mathematics in general... For example, now with Grothendieck's reinterpretation of the word point, and particularly with results of Bhargava that most hyperelliptic curves over Q have no rational points, I could certainly imagine a philosophically mature discussion of the ontological meaning of "curve" and "point", isolating just what it is that makes a curve a curve and that makes a point a point, and in what sense either exist. Perhaps that could clarify mathematical thinking. – Daniel Moskovich Oct 2 '14 at 12:41
• @DanielMoskovich : All that can certainly be done. However, the way you phrased your question initially made it sound like it made a big difference to you whether the people carrying out this discussion were employed by the Department of Philosophy or the Department of Mathematics. I personally think that the employment question is not so interesting, but if one insists on pigeonholing people as either "philosophers" or "mathematicians" in this way, then my point is that socioeconomic factors make it likely that the discussants will be "mathematicians." – Timothy Chow Oct 2 '14 at 18:34
• Your examples of "curve" and "point" are good ones because a common stumbling block for students of modern algebraic geometry is a preconceived notion of a "curve" as a "set of points." It takes a mental adjustment to think of a curve as an abstract entity that is "defined over" one field but may have "$F$-rational points" for various fields $F$. This shift in viewpoint could be called "philosophical" and there's no reason a "philosopher" couldn't elucidate it beautifully, but for sociological reasons it will usually be a "mathematician" who is tasked with explaining this type of "philosophy." – Timothy Chow Oct 2 '14 at 18:46
• I think you have a point, but I feel it applies mostly to philosophy. Math, and many other disciplines, seem to have an independent definition that is quite congruent with what the academic department of that name does, but I don't know a definition of philosophy that makes it clear that ethics, epistemology and the history of science are all parts of a coherent whole. It also seems that you do not have to be trained in academic philosophy or otherwise endorsed by a philosophy department in order to do philosophically-oriented math or physics, at least for some definitions of 'philosophical'. – sdenham Feb 28 '17 at 20:57
• @sdenham : Philosophy does have fuzzier boundaries than most disciplines, but the boundaries of math are not completely sharp either. If you look at universities around the world, you'll see that subjects such as applied math, statistics, computer science, operations research, and mathematical physics are sometimes considered part of the same subject and sometimes are not. Also, logic is sometimes considered math but is sometimes considered philosophy. – Timothy Chow Mar 1 '17 at 0:23

Turing has already been mentioned in previous excellent answers as someone whose ideas sit at the boundary of philosophy and mathematics, conventionally understood. I want to mention Ludwig Wittgenstein in this context as an example of someone grappling with a host of similar ideas but who arguably took a more "philosophical" approach to them, one that explicitly resists formalization.

The book "Wittgenstein's Lectures on the Foundations of Mathematics, Cambridge, 1939" is a transcript of a seminar given by Wittgenstein which was attended by Alan Turing, among a handful of other notable young Cambridge scholars of the day. It contains an interesting exchange between Turing and Wittgenstein about what happens when people disagree as to the result of a calculation. Turing insists that some are right and some are wrong, and Wittgenstein wonders what that might mean. Turing says that if you build a bridge that depends on a wrong calculation, it will fall down. My recollection is that Wittgenstein argues that this doesn't cause the bridge to fall down, but that the bridge falling down might serve as a definition of what it means for the calculation to be wrong.

I gather that many people find Wittgenstein to be obtuse and/or just plain confused, but a more charitable reading suggests that he was wrestling with ideas related to undecidability, albeit from a much broader sociological vantage point. See, for example, "A note on Wittgenstein's notorious paragraph on the Gödel theorem". If a bridge standing or falling down depended on a proposition that was undecidable, what then? I believe, but am not sure, that the issue of "in which formal system" would not necessarily have been in the air at the time.

So, on the one hand we have a nice example of philosophical questions stimulating what went on to become a much more formal (and elaborate) theory in the work of Turing. On the other hand, I feel that the hypothetical scenario about the bridge is a "purely" philosophical question that is both interesting and challenging and that is not addressed by the subsequent formal developments. Naively put, which formal system does mother nature obey? Moreover, how can we make better sense of this question? Philosophy gropes at such questions; once they have been sufficiently sharpened, mathematics constitutes the work of refining and extending our understanding.

• To use a more contemporary example, instead of a bridge depending on an independent statement, one might consider that someone designs an internet security system based on the assumption $P\neq NP$... – Joel David Hamkins Oct 1 '14 at 21:54
• I don't think I agree with the interpretation of Wittgenstein's argument as having to do with undecidability. Wittgenstein' criticism is more fundamental (and perhaps therefore less interesting from a mathematical point of view): take some proposition perfectly decidable, with a complete and valid proof in a given systems of axioms. What do you do if someone tells you: "I understand the system of axioms, but I don't think this proof proves the proposition according to the rules we agreed on". Well, surely he must have misunderstand these rules, but how do we ensure that he and me... – Joël Oct 4 '14 at 18:33
• ... understand the same rules in the same way? What I mean is that Wittgenstein puts into question not only the feasibility of Hilbert's finitist program of fundations in mathematics (as Gödel's theorem) does, but the very meaning of such a program. – Joël Oct 4 '14 at 18:36
• @Joël Sure, W's arguments are broad and (perhaps intentionally) informal--I don't mean to suggest that he was thinking exclusively or explicitly about undecidability. Rather, I feel that certain aspects of his thought admit to formalization (the preoccupation with rule-following, which can lead to a theory of computability), while other parts (the question of how our physical world might instantiate certain rules) are more purely philosophical. I'm no expert on Wittgenstein or computability, but wanted to throw W's name into the ring anyway. Thanks for chiming in. – R Hahn Oct 4 '14 at 20:44

George Berkeley’s „The Analyst” is an example.

Berkeley was a philosopher. He had a philosophical motivation.

He attacked a mathematical concept of “Infinitesimals”. He clarified that an original approach to Infinitesimals was not rigorous.

It contributed to the development of epsilon-delta definition and non-standard analysis.

I can see an essential difference between Pythagoreans being against irrational numbers and Berkeley criticizing infinitesimals. The Pythagoreans were using metaphysical arguments and it was against the development of mathematics. The contribution of Berkeley who spotted gaps was positive.

• A review you can read here of Maclaurin's collected letters does not support the idea that Berkeley spurred any development in analysis. And I am not aware that Weierstrass was influenced by Berkeley either. – Marius Kempe Oct 1 '14 at 13:22
• For sure, Berkeley spurred at least "An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst" by Bayes. The whole issue seems debatable. Robinson wrote “The vigorous attack directed by Berkeley against the foundations of the Calculus in the forms then proposed is, in the first place, a brilliant exposure of their logical inconsistencies.” Nevertheless, I'm not a historian and maybe I simply fell prey to a widespread myth. – Waldemar Oct 1 '14 at 13:47
• I agree with Waldemar. Berkeley did not influence Weierstrass directly, but he put in the minds of all mathematicians the idea that there was a foundational problem in the work of Newton and Leibniz. Without him, most lesser mathematicians (that is, almost all) would have wrongly believed that whatever difficulties they might have with foundations of calculus were due to their own limitations. Berkeley spurred a desire for laying down strong foundations for calculus for generations of mathematicians, and many tried with no or only partial success before the final push of Bolzano & Weierstrass – Joël Oct 1 '14 at 20:44
• @MariusKempe: I'm in the early stages of reading Fowler's book, but am not yet sure what to make of it. I'm sure my understanding will change once I've read more. I would guess that if Fowler's thesis is correct, the Pythagorean discovery that the diagonal of the unit square has irrational length could not have happened in that way because Greek mathematicians never thought of length arithmetically in the first place. Nevertheless, it is clear that mathematicians of the Pythagorean School and... – Will Orrick Oct 2 '14 at 18:45
• ... of Plato's Academy were grappling with the issues raised by incommensurability (in the context of geometry, if not arithmetic). Eudoxus was a student of Archytas, a Pythagorean, and of Plato, heavily influenced by Pythagoreanism. Whether the foundational crisis occurred or not, it doesn't seem that Pythagorean philosophical prejudices hindered the development of mathematics. – Will Orrick Oct 2 '14 at 18:46

Per Martin-Löf has done work spanning mathematics, logic, and philosophy. I am not sure how much his early work on randomness was influenced by philosophical concerns, but his work on type theory in the 70s and 80s certainly was. This is especially clear in the 1984 lecture series, On the Meanings of the Logical Constants and the Justifications of the Logical Laws, which was highly influential in computer science.

• Without Martin-Löf's type theory, fascinating developments like homotopy type theory seem unconceivable. – Manuel Bärenz Jun 27 '18 at 9:16

Just like René Descartes' La Géométrie from Discours de la méthode (1637) might have had a profound influence on mathematics, also Immanuel Kant's Transzendentale Ästhetik from Kritik der reinen Vernunft (1781) might have had a profound influence on mathematics.

In his Habilitationsvortrag (1854), Bernhard Riemann talked about his success in the task given by Carl Friedrich Gauß to "update" Kant's intuition in the face of new facts like non-Euclidean geometry. When this talk (and a sketch of the elaboration of the "proposal") was published in 1867 as
Ueber die Hypothesen, welche der Geometrie zu Grunde liegen
On the Hypotheses which lie at the Bases of Geometry
it lead to responses like the one from Hermann Helmholtz in 1868
Ueber die Thatsachen, die der Geometrie zum Grunde liegen
The Origin and Meaning of Geometrical Axioms
which in the end also influenced Sophus Lie's On a class of geometric transformations (1871), which in turn also had an influence of Felix Klein's Erlanger Programm (1872).

Well, maybe the influence of Kant or Riemann or Helmholtz or Lie, or even Klein wasn't such important, but still these developments were always responses to earlier published philosophical positions.

• This comes to mind when you say Kant. – Asaf Karagila Oct 7 '14 at 10:47
• @AsafKaragila Nice. Maybe Kant caused mischief to philosophy, but the mischief to mankind probably had another source: language! What en.wikipedia.org/wiki/Standard_German describes is called Hochdeutsch, and it was an intentional invention. It had a similar purpose like the "Euro" currency in current times. One of the intentions was to create "reality" (the one they wished for and knew they wouldn't get), and that's just what it did (create and destroy more "reality" than its inventors could ever imagine)... – Thomas Klimpel Oct 7 '14 at 11:27

There is a book that gained a cult following in the 1970s, and this book lies at the intersection of philosophy and mathematics. It's Laws of Form by G Spencer Brown.

Few serious mathematicians have held it in high regard, and little new creative work can be traced to it. And it's a little on the abstract side, when compared to most works on philosophy. But it can be an interesting way to look at logic and arithmetic as two different expressions built on top of a single concept.

Brown calls this the calculus of distinctions. Making a distinction may be the most fundamental thought process a child first learns.

• Although it has struck the fancy of some mathematicians who have responded creatively. Louis Kauffman comes to mind. – Todd Trimble Oct 1 '14 at 11:06
• Good point. Maybe I should have said "some serious mathematicians". – Walter Mitty Oct 1 '14 at 11:46
• I find this a fantastic example, which I should have thought of before asking the question. Especially given that I also found it very useful and it stimulated my own thinking... – Daniel Moskovich Oct 1 '14 at 13:20

Lawvere has evidently applied a philosophical outlook to some effect in his mathematics.

This answer by Urs Schreiber on philosophy stack exchange gets at this and more: https://philosophy.stackexchange.com/a/9814

I am very skeptical of a philosopher talking about Math, given the unavoidable interpretative elasticity of the philosophical discourse and the blunders of some of its practitioners. (Hegel's name comes to mind.) In particular, I don't find it productive to learn Math from a philosopher, or to turn to philosophy when your research is stuck hoping to find salvation there. It looks to me that philosophers' discussions and debates come after major advances in sciences, and not the other way around. Philosophers may claim that Turing is one of their own. However, he is famous for his mathematical contributions.

• As you know, he had a degree in math, a PhD in math, and his work on computability was prompted by questions formulated by another famous mathematician David Hilbert. Turing's famous 1937 paper was published in the Proceedings of the London Mathematical Society, which should tell you that he thought of his contributions as being mathematical – Liviu Nicolaescu Oct 1 '14 at 9:36
• In fact, maybe you should have a chat sometime with for example David Corfield or Steve Awodey or Colin McLarty, all three of whom are professional philosophers. Each of them has made significant contributions to mathematics (the latter two publish in mathematics, and DC is an important presence in his activities and creative responses at the nLab and nForum which reflect deep mathematical insight). – Todd Trimble Oct 1 '14 at 11:28
• Well, interpretations of what Hegel was driving at will of course differ in their charitableness. However, my main point is to speak to "I am very skeptical of a philosopher talking about Math" because my own experience is that some of them are engaged in deep mathematics. I gave a few contemporary examples. – Todd Trimble Oct 1 '14 at 12:27
• Todd Trimble: But these are examples of a philosopher doing mathematics, not philosophy, just like C.S. Peirce. – Daniel Moskovich Oct 1 '14 at 13:22
• @DanielMoskovich I won't argue this further, except to repeat that I take strong exception to the specific announcement "I am very skeptical of a philosopher talking about Math" -- that sounds like a prejudice that ought to be redressed. My point here is that some philosophers when talking about Math know very well what they are talking about -- even when they are doing philosophy of mathematics. (For extra clarity: I wasn't attempting to give an answer to the Original Post; I was merely commenting on something Liviu wrote.) – Todd Trimble Oct 1 '14 at 16:43

Logic (and formal systems) is a contribution of philosophy.

The point is that up to recent times (~1800), mathematics was a discipline of philosophy, just as music was a discipline of mathematics at the same level as arithmetics, geometry and astronomy. It was impossible for a mathematician not to study philosophy.

So you question is meaningless up to 1800. Just read Leibniz or Newton to see that the question of how god made the universe was a mandatory base to justify axioms, which we nowadays consider as hypothesis.

As far as modern science is concerned, philosophy is a science of human ideas and beliefs which escape to any mathematical models, and will probably ever be.

During the transition period (1800-1900), lots of work has been done on the common part of mathematics and philosophy, which resulted in a formalisation of logic and a pedestal to base mathematics.

• You say that because mathematics was part of philosophy, it was impossible to study mathematics without studying philosophy, and that music was part of mathematics. Are you saying that it was impossible for a musician not to study mathematics, too? Exactly what mathematics was studied by musicians and how did they study it? – Douglas Zare Jun 17 '16 at 2:56

Perhaps it has been cited elsewhere in this context, but Hilary Putnam's 1980 short essay "Models and reality" brings together mathematics and philosophy in its discussion of $V = L$ and the Loewenheim-Skolem theorem: http://www.princeton.edu/~hhalvors/teaching/phi520_f2012/putnam1980.pdf

Another source of recent interaction is the field of axiomatic theories of truth (Kripke, Herzberger, Leigh and Rathjen and others). For example: http://www1.maths.leeds.ac.uk/~rathjen/AbsoluteEnd_Truth.pdf

Maybe these are examples of mathematical logic clarifying philosophy...

Would you expect to learn much about how to practice science (or any sub-branch thereof) from the philosophy of science? I hope the answer would be no; that falls within the study of science.

The philosophy of science and mathematics exists to examine the philosophical underpinnings of those disciplines which lie outside the purview of those disciplines themselves (e.g. do numbers have some kind of objective existence separate from the minds of reasoning beings), and philosophical topics related to them (such as the ethics of scientific and mathematical enquiry).

Now, that is not to say that this stuff is useless - it can guide the actions of scientists and mathematicians, and identify limitations in the discipline and its methods. However, you're not going to find material which has a simple relationship to the practice of the discipline, and certainly not in introductory texts aimed at people who are not mathematicians.

• That's fine- philosophical underpinnings can clarify. I'd feel greatly enlightened by a mature philosophical treatment of, for example, quantum invariants of knots and 3 manifolds. Or just of knots and of 3-manifolds. Or of curves, or of points. I am really not seeing that though. – Daniel Moskovich Oct 4 '14 at 17:13

Ludwig Wittgenstein invented the truth table, in trying to clarify the structure of propositional logic. Of course, this is now a standard tool in mathematics and computer science. Wittgenstein was trained as an engineer and working as a philosopher at the time, but was never a mathematician.

• Truth tables in all but the usual graphical form were around long before Wittgenstein, going back to Buridan and Ockam. The philosophy of Buridan, Ockam, Russell (pre-1940's), Wittgenstein, etc is nominalism, the belief in contingency of all truths (atomic/brute facts, facts without further reason), hence properly criticized by Blanshard and Goedel as the arbitrary creation view. Truth table arguments were naturally found there. Many famous logicians of the 19th century (all of them mathematicians) also used what are basically truth table arguments but without spurious philosophical baggage. – Gottfried William Oct 4 '14 at 20:37
• I agree that Wittgenstein did not invent the truth tables. However, it should be added that 19th-century logicians were arguing about the character and purpose of symbolic logic putting a lot of philosophical baggage into their arguments. Frege in his Begriffsschift opposed Boole, Peirce and Schroeder. See ontology.co/two-views-language.htm for descriptions of these arguments by J. Hintikka and J. van Heijenoort. – Margaret Friedland Oct 4 '14 at 22:43

To add to the discussion: the very dichotomy Philosophy x Mathematics didn't make sense to most thinkers of the past. That still holds true for many of today's philosophers - Philosophy is not a discipline, like Mathematics or Chemistry.

That being said, I could list tens of philosophers that satisfy your expectations: Frege, Russell, Tarski, Smullian; the list goes on and on. Now, for those that don't satisfy your expectations, but may be even more interesting to you for that exact reason, I'll mention Husserl, Wittgenstein and classics like Pythagoras and Plato (yes, they had a lot to say about "Mathematics" and ontology).

Based on this sentence alone: "Philosophy might elucidate the 'true meaning' of axioms and of definitions by examining their ontology in a wider context", I would definitely give Wittgenstein a try. You'd be surprised to know that the sun may not rise tomorrow...

I'm surprised no one has yet mentioned Cantor. His work leading to Transfinite Arithmetic, and from there to general Set Theory, overturned millenia of received opinion on the concept of the infinite by insisting on the mathematical existence of actual, completed infinities. He defended this work with a body of philosophical arguments arguing that the earlier conclusion - that infinity could of necessity only be potential - was in fact mistaken. [source: biography of Cantor by (I forget who, sorry)]

• Rubbish. Leibniz, Euler, and others used infinite numbers galore without awaiting Cantor's permission. – Mikhail Katz Jun 16 '16 at 15:35
• @MikhailKatz: I don't think that Leibniz and Euler used completed infinities as opposed to potential infinities. While the distinction between these two types of infinity is not used in mathematics today, it was certainly used in the past, and I think it's fair to say that Cantor was the first really successful advocate (in mathematics) of completed infinities. – Timothy Chow Jun 16 '16 at 20:23
• @PMar : Lennart Meier mentioned Cantor. – Timothy Chow Jun 16 '16 at 20:24
• @TimothyChow, my comment was prompted by PMar's reference to "concept of the infinite". The infinite does not necessarily have to be channeled into set theory, and the founders of the calculus made ample use of it though they did not express their insights in either naive set theory or ZFC. I agree with your point about their attitude toward infinite collections. – Mikhail Katz Jun 17 '16 at 7:34
• Perhaps the Cantor biography PMar has in mind is Joseph Warren Dauben, Georg Cantor. His mathematics and philosophy of the infinite. Princeton University Press, Princeton, NJ, 1990. xiv+404 pp. ISBN: 0-691-02447-2 MR1082146 (91h:01044) – Gerry Myerson Feb 28 '17 at 22:22

After Newton and Leibniz developed infinitesimal calculus, a number of critics emerged to criticize the new technique. These included Berkeley, Cantor, and others. One of the few schools to battle the negative attitude toward infinitesimals at the beginning of the 20th century was Hermann Cohen's school, also known as Marburg neo-Kantianism. Their efforts did not bear fruit immediately but they did influence one Adolf Fraenkel, who passed on his interest in infinitesimals to one Abraham Robinson. In his autobiographical work in the 1960s Fraenkel noted that "Robinson saved the honor of infinitesimals."

Thus Cohen's attempts to develop a respectable theoretical basis for a philosophy of infinitesimals eventually bore fruit in Robinson's framework for analysis with infinitesimals.

• William Lawvere, another philosophically-minded mathematician, also developed another elegant approach to infinitesimals: synthetic differential geometry. – ಠ_ಠ Oct 8 '17 at 22:06
• I believe the famous quote from The Analyst: a Discourse addressed to an Infidel Mathematician by George Berkeley (1734) is appropriate here: "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" – მამუკა ჯიბლაძე Oct 9 '17 at 3:58
• @მამუკაჯიბლაძე, the phrase you mentioned is out of date. What is more relevant today is the observation that historians of mathematics are involved in a quest for the ghosts of departed quantifiers. Namely, they seek to interpret the work of Leibniz, Euler, Cauchy, and other pioneers of infinitesimal analysis in such a way as to replace their use of infinitesimals by long-winded quantified paraphrases. Related literature can be consulted here. – Mikhail Katz Oct 10 '17 at 8:33
• Yes, the synthetic approach is cool. Still, another famous quote comes to my mind - "The method of "postulating" what we want has many advantages; they are the same as the advantages of theft over honest toil." (Bertrand Russell, Introduction to Mathematical Philosophy (1919)). I would not mind that in fact, but what raises my suspicions is that regarding the most interesting synthesized entities it is still not known whether their existence leads to contradiction or not. And what to do if it is consistent but is not compatible with, say, ZFC. – მამუკა ჯიბლაძე Oct 10 '17 at 14:20
• @ಠ_ಠ, an approach to synthetic differential geometry via Robinson's framework can be found here. – Mikhail Katz Oct 13 '17 at 8:55

Yes, philosophy has clarified mathematics.

In 'Naming Infinity', by Loren Graham and Jean-Michel Kantor, they argue that Russian mathematicians in the early 1900s were able to introduce new concepts to set theory due to their philosophical (and religious) perspective on free will and naming things.

"At the core of this book is the contest between French and Russian mathematicians who sought new answers to one of the oldest puzzles in math: the nature of infinity. The French school chased rationalist solutions. The Russian mathematicians, notably Dmitri Egorov and Nikolai Luzin—who founded the famous Moscow School of Mathematics—were inspired by mystical insights attained during Name Worshipping. Their religious practice appears to have opened to them visions into the infinite—and led to the founding of descriptive set theory."

I think this is a clear example of how our approach to mathematics depends on our philosophical beliefs. Both the French and the Russian mathematicians saw mathematics as a language that describes reality. For the French, this prevented them from advancing in set theory, since it ran against their beliefs of a continuous and orderly world. Meanwhile, the Russians were able to solve the contradictions in set theory and pave the way for new mathematical disciplines. Egorov and Luzin thought free will and the power of names were universal truths that existed in every discipline, not just philosophy or religion. They saw discontinuous functions as a way to describe freedom of choice. Ultimately, their perspective brought about a lot of progress in mathematics.

• What's the consensus on the accuracy/reliability of Graham and Kantor's central thesis? I've read mixed reviews. Also, doesn't this contrast-the-French-with-the-Russians framework have gaps when it comes to e.g. Hausdorff or Sierpinski or Kuratowski? – Yemon Choi Mar 1 '17 at 1:59
• @Yemon: Between France and Russia there are Germany and Poland. It's plain geography... :-P – Asaf Karagila Mar 1 '17 at 6:12

Bernhard Riemann's famous 1854 Habilitation lecture "On the hypotheses…" owed more to the influence of the anti-Kantian philosopher Johann Friedrich Herbart than to Kant himself. Here one could mention an illuminating article by Nowak:

G. Nowak, in The history of modern mathematics, Vol. I (Poughkeepsie, NY, 1989), 17–46, Academic Press, Boston, MA, 1989.

Nowak argues that

1. Herbart's constructive approach to space, already cited, mirrored the content of Riemann's reference to Gauss in that both discussed construction of spaces rather than construction in space.

2. Riemann followed Herbart in rejecting Kant's view of space as an a priori category of thought, instead seeing space as a concept which possessed properties and was capable of change and variation. Riemann copied some passages from Herbart on this subject, and the Fragmente philosophischen Inhalts included in his published works contain a passage in which Riemann cites Herbart as demonstrating the falsity of Kant's view.

3. Riemann took from Herbart the view that the construction of spatial objects was possible in intuition and independent of our perceptions in physical space. Riemann extended this idea to allow for the possibility that these spaces would not obey the axioms of Euclidean geometry. We know from Riemann's notes on Herbart that he read Herbart's Psychologie als Wissenschaft.

It is possible that, had Kant's ideas about infinite space not been challenged by Herbart, the field we know today as Riemannian geometry may have developed much later.

This is meant to complement Thomas Klimpel's answer on Riemann.

Note. Peter Heinig pointed out that the historian David Rowe similarly holds that Riemann was more of a Herbartian than Kantian; see comments below and also Rowe's review.

• It's been a while since I read it, but I think Jeremy Gray's book "Plato's Ghost" goes into Riemann's (and others') philosophical motivations in some detail. – Noah Schweber Oct 15 '17 at 15:11
• "This is meant to complement Thomas Klimpel's comments on Riemann above." For some value of "above". – Gerry Myerson Oct 15 '17 at 21:45
• I was referring to this answer @GerryMyerson – Mikhail Katz Oct 16 '17 at 6:58
• Yes, that answer is about two feet below your answer (on my computer screen, at the present time). – Gerry Myerson Oct 16 '17 at 8:37
• This very interesting answer is slightly damaged by a probably accidental misuse of the English word 'concept', in "instead seeing space as a concept which possessed properties and was capable of change": the use of 'concept' here runs counter to the 'instead'; a 'concept' cannot "possess properties"; I think the 'mot juste' here would be 'substance' (reference: [...] Jill Vance Buroker: Space and Incongruence: The Origin of Kant's Idealism, Springer 1981, where one reads "Newton's theory is unintelligible because he conceives of space as a non-substance with properties. But the [...] – Peter Heinig Oct 16 '17 at 11:16

There's a great book called the "Synthetic Philosophy of Contemporary Mathematics" by Prof. Fernando Zalamea that came out a few years ago. I've read about half and would highly recommend.

It's intention is not so much to clarify mathematics for mathematicians but to clarify contemporary advances in mathematics to mathematical philopsophers. In a sense, then, it distills and analogizes contemporary mathematics into "intuitive" nuggets that can be used in a philosophical sense.

I would recommend. There are also some interesting discussions on the n-category café

https://www.amazon.com/Synthetic-Philosophy-Contemporary-Mathematics-Fernando/dp/0956775012

Furthermore, work done by Ricardo and David Nirenberg on these aspects, e.g.: http://criticalinquiry.uchicago.edu/uploads/pdf/nirenbergs_badiousnumber_complete.pdf

Kant has featured in Geog Cantor's argument that time is irrelevant for describing the continuum.

This is a partly a (tangentially) relevant answer to the question, and partly a relevant comment on a very interesting comment at 'Asaf Karagila Oct 7 '14 at 10:47' in this thread, which in effect points out that Cantor considered Kant harmful and a bad mathematician.

I just stumbled over an 1883 passage of Cantor's in which a thirty-years-younger (than the Cantor who wrote the 1911 letter to Russell, that is) has Kant have sort-of a cameo appearance in what arguably is transfinite-set-theory's foundational-document: in [G. Cantor: Grundlagen einer allgemeinen Mannichfaltigkeitslehre. Teubner 1883, p. 29] one finds:

[my translation:]

First of all, I have to declare the following: in my opinion it is not correct to adduce a concept of time or of a (intuitive-)view-on-time [translated intentionally non-idiomatically] if one explains the much more original and general concept of the continuum; I think that time is an idea which, in order to be explained clearly, presupposes the the concept-of-the-continuum, which is independent of time; also, even with the help of the concept-of-the-continuum, [time] can neither be objectively conceived of as a substance, nor subjectively conceived of as a necessary apriori form-of-intuition, rather time is nothing other than an auxiliary and relative concept, by which one determines the relation between distinct movements which occur in nature and are perceived by us. [Some would argue that Cantor is anticipating both Einstein and Barbour here.] Something like objective or absolute time nowhere occurs in nature and therefore one also cannot view time as a measure of movement, rather, to the contrary: movement could be viewed as a measure of time, were it not for the fact that, which contradicts the latter interpretation, that time, even in its modest role of a subjectively necessary apriori form-of-intuition, has failed to reach any fruitful, unchallenged state of prosperity [Cantor gets rather ornate at this point, and, in a sense, weakens his own case by negating an extreme condition ('time has not reached an unchallenged etc etc'; what wonder.)], even though it has had enough time to do so since K a n t.

In short, Cantor uses a reference to Kant's influential philosophy to strengthen his argument that time is irrelevant for explaining the continuum.

This can be seen as an example that "philosophy" "clarified" "mathematics".

Of course (thanks to Asaf Karagila for pointing out that this was not made clear in this post), one can interpret Cantor as writing rather ironically about Kant here, proposing something like the following rhetorical question (I am paraphrasing):

'Now if what Kant wrote were so true, how come that a hundred years later [which is roughly the time span that had passed when Cantor wrote this][which is roughly the time span that had passed when Cantor wrote this] the view on 'time' as a 'subjective necessary apriori form-of-intuition', this point-of-view still has not taken hold?'

Now one can argue that Cantor's argumentation itself is a fallacy, or a rather weak charge against Kant, and by itself does not speak against Kant (let alone contradict him), since of course *it does not necessarily imply the wrongness of a view that a view is not widely held*.

This post is not to 'endorse' either of these interpretations, rather to point out a relevant and historical interesting aspect of Cantor's work.

The present observation can be seen as philosophy having clarified mathematics, if only in the indirect way that (0) Cantor seems to have disliked Kant, whether at age 37 or at age 66, and therefore (1) distanced himself from, and avoided, the use of any notion of 'time' in his mathematics.

• Didn't Cantor say that we should stop listening to Kant, especially about mathematics, of which Kant knew very little about. Or am I conflating the people involved in this quote? – Asaf Karagila Oct 8 '17 at 21:40
• @AsafKaragila: thanks for pointing out. You are right in saying that Cantor, even in 1882, seems not be 'in favor' of Kant in any way; the passage here can rather be interpreted as referring to Kant ironically, something like 'if what Kant wrote were so true, how come that a hundred years later ${}_{\text{[which is roughly the time span that had passed when Cantor wrote this]}}$ the view on 'time' as a 'subjective necessary apriori form-of-intuition', this point-of-view still has not taken hold?' I was aware of that, but did not make it clear enough, and will edit accordingly. Thanks. – Peter Heinig Oct 9 '17 at 5:52
• Huh. I haven't noticed that you mentioned my comment from before; nor I remembered making it. Ha! :) – Asaf Karagila Oct 9 '17 at 8:31

I'm not sure that some people would agree... but Popper's work on provability has clarified how things should work in "classical" statistics, that we can only prove something is wrong not that it is right - black swans anybody? We can only say that we can/cannot reject the null hypothesis... whereas most people want to (incorrectly) accept the null/alternative hypothesis.

• Do you mean provability in mathematics or in real life? Your answer seems (to me at least) to be more about extending the mathematical concept of a proof outside mathematics than mathematics itself. – Joonas Ilmavirta Oct 5 '14 at 8:45
• Popper's idea is commonly referred to as "falsifiability". A more recent development of Popper's ideas can be found in the work of Deborah Mayo in "Error and the Growth of Experimental Knowledge". Her works seeks to provide philosophical justification for the widespread use of formal statistical testing methods in scientific practice. – R Hahn Oct 6 '14 at 15:36

I paraphrase with italics from the original post to give my perspective:

My naïve expectation would be that metamathematicians * might take a difficult construction or proof, and clarify it by isolating the key ideas behind it. Having isolated the key ideas, they might then highlight their relevance and thus point the way forward. Beyond this, I would hope that such practitioners might elucidate the true meaning' of axioms and of definitions by examining their ontology in a wider context.

I first took meta-mathematics hoping to discuss things like the duality between the notions of points and lines in axiomatic geometry, so that one could find ways to discover and prove more results by exploiting the symmetry of the foundations of a given theory. I found something completely different, of course, with only occasional references to duality or n-ality. I put * after metamathematicians, because, after some years seeing how some good mathematicians practiced (they would not stop at a single result: they would poke it, prod it, extend it this way or that, try to find limits by tweaking some aspect of the assumptions or proof and come up with counterexamples, and use other methods to look for wideness of application, or anticipate when such attempts might fail), I realize now that the first term should be "professional mathematicians", or just "mathematicians" perhaps.

(This is not to dismiss philosophy from the question, so much as to ask for an alternate basis for the question. I think philosophical examination might yield guidance as to which logical system to use in certain pursuits, but it would still depend on the naive expectation above being carried out by mathematicians.)

Gerhard "Mathematicians Know What 'Is' Is" Paseman, 2014.10.01

Can the works of Penelope Maddy ("Believing the Axioms", I and II), Georg Kreisel (his papers on "Informal Rigour"), and Benjamin Rin's paper "Transfinite recursion and the iterative concept of set" (to name just a few), be said to have not elucidated, or clarified mathematical concepts, proofs, or constructions in a way that was useful to research mathematicians?