Contemporary philosophy of mathematics

Question

Starting to write an introduction to the philosophy of mathematics, I find tons of positions that are of historical interest. Which philosophical positions are explicitly considered these days, say in the last ten years?

Answer 1

Let me mention a few current issues on which I have been involved in the philosophy of mathematics. Of course there are also many other issues on which people are working.

Debate on pluralism. First, there is currently a lively or indeed raging debate on the issue of pluralism in the philosophy of set theory. If one takes set theory as a foundational theory, in the sense that essentially every mathematical argument or construction can be viewed as taking place or modeled within set theory (whether or not it could also be represented in other foundational theories), then the question arises whether set-theoretic questions have determinate answers. On the singularist or universist view, every set-theoretic question has a final, determinate truth value in the one true set-theoretic universe, the Platonic realm of set theory. On the pluralist or multiverse views, we have different conceptions of set giving rise to different set-theoretic truths. Both views are a form of realism, and so the debate breaks apart the question of realism or Platonism from the question of the uniqueness of the intended interpretation.

I discuss these issues at length in my paper

• Hamkins, Joel David, The set-theoretic multiverse, Rev. Symb. Log. 5, No. 3, 416-449 (2012). DOI:10.1017/S1755020311000359, ZBL1260.03103.

There is a growing literature discussing these issues, part of which you can find here. See also the multiverse tag on my blog.

Potentialism. Another currently active topic of research is the issue of potentialism. This topic arises classically in the idea of potential versus actual infinity, with quite a long history, but current work is looking into various aspects of the classical debate, particularly with respect to considering potentialism as a modal theory.

That is, one separates the potentialist idea from the issue of infinity, and looks upon the potentialism as concerned with the idea of having a family of partial universes, or universe fragments, which can be extended to one another as the universe unfolds.

Øystein Linnebo has emphasized this modal nature to potentialism, and he and I undertook to analyze the precise modal commitments of various kinds of set-theoretic potentialism in our paper:

• Joel David Hamkins and Øystein Linnebo, The modal logic of set-theoretic potentialism and the potentialist maximality principles, arXiv:1708.01644.

See also the slides for the tutorial lectures series I gave recently on Set-theoretic potentialism, Winter School in Abstract Analysis 2018.

For an example, I believe that the use of Grothendieck-Zermelo universes in category theory exemplifies the potentialist outlook, since one works inside a given universe until a need arises for a larger universe concept, in which case one freely moves to the larger universe concept.

My work on arithmetic potentialism in

• Joel David Hamkins, The modal logic of arithmetic potentialism and the universal algorithm, arxiv:1801.04599.

provides a way to understand the philosophy of ultrafinitism, viewing it ultimately as a form a potentialism.

Answer 2

I've written fairly extensively on predicativism and on the paradoxes of set theory and logic. My central claims are (1) it makes better sense, both mathematically and philosophically, to regard all uncountable collections as proper classes, and (2) intuitionistic logic can be used to definitively resolve the classical paradoxes.

I've written one book, Truth & Assertibility, and I have a bunch of papers on the arXiv:

Predicativity beyond $\Gamma_0$

Analysis in $J_2$

Mathematical conceptualism

Axiomatizing mathematical conceptualism in third order arithmetic

The concept of a set

Is set theory indispensable?

Constructive truth and circularity

Intuitionism and the liar paradox

Kinds of concepts

The semantic conception of proof

Truth and the liar paradox

Reasoning about constructive concepts

Paradoxes of rational agency and formal systems that verify their own soundness

The liar paradox is a real problem

Answer 3

It's not entirely clear to me what you mean by a "position." Under the most obvious interpretation, things like Platonism, logicism, formalism, intuitionism, finitism, etc., are "positions." However, it seems you are attempting to probe the community of practicing research mathematicians for the "positions" that people currently hold. My anecdotal sense is that the majority of working mathematicians do not have a clearly articulated "position" that they adhere to, since they're mostly busy doing mathematics, and doing mathematics typically does not require taking any of the above "positions." (People who work in areas such as logic and set theory are an exception, since they typically have thought hard about such questions, whether or not they publish anything that is explicitly philosophical, but they are in the minority.)

An analogy might be quantum mechanics. There are many interpretations of quantum mechanics, but many physicists do not commit themselves to any particular interpretation. What matters for their research is that they can do the relevant calculations when necessary, and doing the calculations and making predictions does not require taking a stand on the "positions" that are hotly debated in the literature on the philosophy of quantum mechanics.

So would consider, "I don't really know and I don't really care" to be a "position"? Or are you only interested in the opinions of people who do care and can articulate and defend a well-defined "position"?

There are some mathematicians who have written books whose primary goal is to introduce the reader to some particular area of mathematics, but that also are motivated by an underlying "position" that is not necessarily explicitly articulated or argued for. As someone else already mentioned, Homotopy Type Theory is one such book, and Conceptual Mathematics is another. These books are not (for example) explicitly "structuralist" or "intuitionistic" but there are overtones of such philosophical ideas in them.

The computer revolution also seems to have fostered a generation of mathematicians with what, in traditional language, might be called a formalist or finitist or even ultrafinitist bent. I'm not aware that this fact has been officially documented anywhere, but you can see it all over MathOverflow. For example, you can see it if you read through the page on Is PA Consistent?.

Ultimately, if you really want to know what the mathematical research community thinks about certain questions, you may need to conduct a formal survey, with carefully constructed questions to find out what you really want to know. If you do that, then you need be careful not to make implicit assumptions that may not be true, e.g., that mathematicians have well-defined stances on questions that philosophers of mathematics are interested in.

Answer 4

Footnote to the nice answers already given. It has not been mentioned yet that over the last decade or so a new standpoint on mathematical ontology has been added to the known ones:

Opinion. $\infty$-groupoids are more fundamental than sets or categories.

This point of view seems to be attributed to V. Voevodsky. I have seen him say something more less equal to the above in at least one talk (where he also related that it took him long to wean himself off the received wisdom that all of mathematics ought to be built from either sets or categories); currently I don't find a reference for the talk. Details on this point of view can be found in more or less every introduction to Homotopy Type Theory. Very briefly, this is about the point of view that the syntax of all of mathematics should be Martin-Löf type theory with some additional axioms, and the intended semantics/ontology ought to consist of $\infty$-groupoids. In a sense, this is the position that the mathematical world consists of $\infty$-groupoids. This can be seen as a new "position" in the philosophy of mathematics that has arisen in the "last ten years", hence is strictly relevant to the OP's question.

Answer 5

Incompleteness, Undecidability, Independence and New Axioms

One major contemporary question in the philosophy of mathematics is "Does (and if so, to what extent) mathematics need new axioms?" This question falls directly out of the work done on undecidability and incompleteness. While Gödel's incompleteness theorems, Turing's theorem of the undecidability of the halting problem, the MRDP theorem, etc. give general examples of incompleteness that can never be "effectively" removed, due to the fact that the theorems are essentially recipes to create undecidable sentences in any given theory that satisfies certain criteria, there are still other statements that are undecidable in certain theories but for which there is hope those statements could be solved by more powerful axioms.

Gödel recognized (see "Remarks before the Princeton Bicentennial Conference on Problems in Mathematics" and "What is Cantor's continuum problem?") that some problems independent of the ZFC axioms of set theory might be able to be resolved on the basis of stronger axioms of infinity and the resulting research effort has been called Gödel's large cardinal program (see Steel "Gödel's Program"). A major portion of contemporary work in set theory is a result of the study of large cardinals. You can find two philosophical discussions of the results (and their limitations in regards to questions such as the continuum hypothesis) by Peter Koellner at Independence and Large Cardinals and Large Cardinals and Determinacy.

This topic has been debated and written about by many mathematical logicians, philosophers of mathematics, as well as contemporary set theorists who do not consider their work to be strictly within the field of logic any longer. An important piece of the debate is contained within the article Does Mathematics Need New Axioms?, which is a collection of four essays and rebuttals on this topic written by Solomon Feferman, Penelope Maddy, John Steel, and Harvey Friedman and presented at the Annual ASL meeting in 2000. From the abstract:

Does mathematics need new axioms? was the second of three plenary panel discussions held at the ASL annual meeting, ASL 2000, in Urbana- Champaign, in June, 2000. Each panelist in turn presented brief opening remarks, followed by a second round for responding to what the others had said; the session concluded with a lively discussion from the floor. The four articles collected here represent reworked and expanded versions of the first two parts of those proceedings, presented in the same order as the speakers appeared at the original panel discussion: Solomon Feferman (pp. 401-413), Penelope Maddy (pp. 413-422), John Steel (pp. 422-433), and Harvey Friedman (pp. 434-446). The work of each author is printed separately, with separate references, but the portions consisting of comments on and replies to others are clearly marked.

Feferman is ultimately skeptical of the program, he believes that there is no hard, pure mathematical reason for new axioms and that therefore the question is only philosophical in nature and thus has no definitive answer. More on Feferman's position (especially on his attitude that the continuum hypothesis is not a definite mathematical problem and is inherently vague) will be referenced further down.

Maddy argues from her naturalist philosophy of mathematics (cf. her book Naturalism in Mathematics for an overview and then especially Maddy's two powerful papers "Believing the Axioms I & II" on the topic of the nature of axioms), that extrinsic justifications for new axioms are not only viable but essential to the development of foundational mathematics.

Steel is of the position that there are still major open questions in set theory that have not been resolved and therefore Gödel's program cannot be shown to be defeated. He argues that there are still outstanding problems in descriptive set theory, as well as problems like the continuum hypothesis, that we have proven independent of ZFC and therefore, of course, we need new axioms to settle them.

And finally, Friedman has taken it upon himself to launch a program in mathematical logic which searches for examples of set theoretic undecidability in finite combinatorics (much more on this later). The justification of needing large cardinal axioms to prove finite combinatorial statements, he argues, is a fairly strong philosophical argument, given that our natural inclination is that these finite statements are indeed true (as in, nobody can doubt them based off of skepticism about infinity). His view can be seen as saying that there are major applications of large cardinals to concrete mathematics, and therefore we cannot throw them aside as purely philosophical and non consequential.

Additionally, in 2011-2012 Peter Koellner organized a project at Harvard titled Exploring the Frontiers of Incompleteness. The goal was:

to bring together some of the most prominent thinkers who have struggled with the following questions:

(1) Do the questions that are independent of the standard axioms admit of determinate answers?

(2) If so then what are those answers and how might we go about determining them?

This subject immediately touches on the pluralism vs. non-pluralism debate that Joel Hamkins has outlined in his answer (of course, he is also a major player in the debate and was part of the project). The multimedia page for the project contains links to preprints, slides, and videos of the lectures of the talks given by each of the participants. Some of the highlights that I think you should consider when going over this topic are:

Hugh Woodin's two papers The Realm of the Infinite and Strong Axioms of Infinity and the search for V. Woodin's work on this area of set theory has shown that in a strong sense, if there is an L like model of a supercompact cardinal, then in the strong sense there is an L like model of every known to be consistent large cardinal. This would give the result that it is possible to form an ultimate $V = L$ like axiom that would resolve many if not all outstanding independence problems, including the continuum hypothesis. There is much in the literature on this topic, I would suggest reading Koellner's response that was included in the project, Woodin on “The Realm of the Infinite”, as well as Colin Rittberg's How Woodin changed his mind: new thoughts on the Continuum Hypothesis.

Feferman was also present and he touched on the same topics as his paper outlined earlier. The highlights from this project are Is the Continuum Hypothesis a definite mathematical problem?, Infinity in mathematics, is Cantor necessary?, The philosophy of mathematics. 5 questions, and What's definite? What's not? as well as Koellner's response Feferman on the Indefiniteness of CH.

Again, Joel Hamkins has provided an answer which outlines his stance in the pluralism vs. non pluralism debate and I agree with his suggestion to follow the links he provided. Here is another link to his The set-theoretic multiverse paper, and Koellner's response from the project Hamkins on the Multiverse.

Charles Parsons outlines the impact that the study of the consistency and relative interpretability hierarchies of formal theories has had on these foundational questions in his paper Evidence and the hierarchy of mathematical theories.

Philip Welch's paper Global Reflection Principles and Koellner's response On Reflection Principles outline the application of reflection principles to the foundation of mathematics and the search for new axioms. To quote the abstract of Koellner's response: "Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified."

Finally the last of my highlights is once again John Steel's work. His four slides The Triple Helix, Gödel's program, Gödel's Legacy, and Generic Absoluteness and the Continuum Problem (I won't link them because they are automatic downloads instead of being hosted on Harvard's server, but they are available on the webpage under his name) give great historical overview of Gödel's large cardinal program, its prominence in contemporary set theory and its lasting philosophical impact.

Finally, I would like to mention a few more points about Harvey Friedman's work. His section of the Does Mathematics Need New Axioms? paper was written during a time when a lot of his work on this subject was first being started. In the almost two decades since, his work has exploded with a plethora of examples of concrete independence of combinatorial statements from ZFC. He wrote an online book titled Boolean Relation Theory, which explains all of the fundamental results of this work. Now, he is currently expanding it into a larger book called Concrete Mathematical Incompleteness which will also discuss the place of incompleteness within logic, mathematics, and philosophy. He wrote an paper for the collection Kurt Gödel and the Foundations of Mathematics Horizons of Truth titled My Forty Years on his Shoulders where he discusses the nature of incompleteness and the impact that Gödel's work has had on mathematics as a whole. In the ending section titled "Incompleteness in the Future" he states:

Mathematics as a professional activity with serious numbers of workers is quite new, let’s say, one hundred years old, although even that is a stretch. Assuming that the human race thrives, what is this compared to, say, a thousand more years? It is probably merely a bunch of simple observations in comparison.

Of course, a thousand years is absolutely nothing in evolutionary or geological time. A more reasonable number is a million years. What does our present mathematics look like compared with mathematics in a million years’ time? These considerations should apply to our present understanding of the Gödel phenomena.

We can, of course, take this even further. A million years’ time is absolutely nothing in astronomical time. Our sun has several billion good years left (although the sun will cause a lot of global warming). Mathematics in a billion years’ time? Who can know what that will be like. However, I am convinced that the Gödel legacy will remain very much alive – at least as long as there is vibrant mathematical activity.

I believe this is a beautiful sentiment which places the concepts of incompleteness, undecidability, independence and new axioms of set theory within their copacetic position in the philosophy of mathematics.

Answer 6

Homotopy Type Theory/Univalent Foundations is probably the newest entrant to the field and should definitely be mentioned. You could do worse than check out the resources at the (quasi-)official site. The HoTT Book has a big introduction, and Mike Shulman and David Corfield have both written either philosophically-inclined, or explicitly philosopical papers on the topic.

Answer 7

Most of the philosophical questions associated with mathematics are usually related to the set theory and logic. However, I am pretty sure there is also a lot to say about the role of computers in the formal proof verification. There is a whole issue of Notices devoted to this topic: Notices Amer. Math. Soc. 55 (2008), no. 11. This issue was, as far as I can tell, inspired by the formal proof of the Kepler conjecture. Perhaps you could contact Thomas Hales who proved Kepler's conjecture and then completed the flyspeck project - formal verification of the proof. He might direct you to suitable sources.

Answer 8

Most of the philosophy of mathematics deals with metamathematics and foundations. But most working mathematicians have only marginal interest to these questions, if any. On my opinion, the principal philosophical question which practicing mathematicians face is this:

How mathematicians choose problems for their research?

What is a good problem, what is a good theorem? How to explain fashion in mathematics? One author who addressed this was G. H. Hardy, "Mathematicians apology". According to Hardy, there are two criteria: a problem/theorem has to be a) non-trivial and b) important. What does this exactly mean he explain with some examples, but on my opinion, there is an enormous potential field of research here. Why some problems attract more attention of mathematicians and others less? Because they are old and difficult? Because some of them come from the so-called "real life"? Everyone can give counterexamples.

Why there is such a huge interest to prime numbers? (Don't tell me that they are important for cryptography: they were not important for ANY real world problem until the middle of 20th century). Why the Greeks were interested in prime numbers but not in the additive number theory (like partitions)?

Why did the Greeks study conic sections in such great detail? (One documented reason is that they found how to double a cube using them, another that perhaps they encountered them when constructing sundials. Do these reasons justify the 12 books of conics of Apollonius which were always compared with Euclid, then and now?)

Why did people struggle for centuries with compass and ruler constructions? For historical reasons? (Then perhaps history of mathematics is one of the most important mathematical subjects of all. Why it is not taught in our universities?)

Why physics is a permanent fertile source of new mathematical ideas while chemistry and biology are not?

Why the theory of dynamical systems after the initial great discoveries of Poincare and Fatou almost fell into oblivion, and had to be revived in 1960-80 to become fashinoable again? One can ask hundreds questions like this.

All these are clearly philosophical questions, with clear applications to the needs of working mathematicians, but philosophers rarely pay any attention to them.

Answer 9

Fernando Zalamea wrote a book in 2009 (english translation in 2013) on contemporary philosophy of mathematics, "Synthetic Philosophy of Contemporary Mathematics" - https://www.urbanomic.com/book/synthetic-philosophy-of-contemporary-mathematics/

(probably the terminology of the book is not very standard for mathematics or for mathematical community, but it's a nice work)

tl;dr There are 5 features specific to modern advanced mathematics mentioned in the book, they are attributed to works of Albert Lautman (from first half of 20th century):

1. a complex hierarchization of diverse mathematical theories, irreducible to one another, relative to intermediary systems of deduction;
2. a richness of models, irreducible to merely linguistic manipulations;
3. a unity of structural methods and conceptual polarities, behind their effective multiplicity;
4. a dynamics of mathematical activity, contrasted between the free and the saturated, attentive to divi­sion and dialectics;
5. a theorematic interlacing of what is multiple on one level with what is one on another, by means of mixtures, ascents and descents.

and 5 other new features mentioned by Zalamea for mathematics after 1950s:

6. the structural impurity of arithmetic (Weil's conjectures, Langlands's program, the theorems of Deligne, Faltings and Wiles, etc.);
7. the systematic geometrization of all environments of mathematics (sheaves, homologies, cobordisms, geometrical logic, etc.);
8. the schematization, and the liberation from set­ theoretical, algebraic and topological restrictions (groupoids, categories, schemas, topoi, motifs, etc.);
9. the fluxion and deformation of the usual boundaries of mathematical structures (nonlinearity, noncom­mutativity, nonelementarity, quantization, etc.);
10. the reflexivity of theories and models onto themselves (classification theory, fixed-point theorems, monstrous models, elementary/nonelementary classes, etc.).

Answer 10

Scott Aaronson's article "Why philosophers should care about computational complexity" might be of interest:

https://www.scottaaronson.com/papers/philos.pdf

Answer 11

The singular universe and the reality of time of Roberto Unger 2015. Section 6 contains discussion of the selective realism philosophy of mathematics which posits that all logical and mathematical statements reside independent of time. I could trace this back to what I would generally call The Gauss Philosophy of Mathematics as exemplified by The General Investigations of Curved Surfaces from 1827.