Clarification on “Reflections on a piece by Frenkel, Part I”

March 24, 2014

During a short exchange the other day on Twitter (tweets 1, 2, 3) Ed Frenkel emphasized something that I regret was unclear in my post here: that he doesn’t prescribe that mathematicians rely on mathematical platonism and what I characterized as its unspecified special epistemological access to knowledge when teaching math. During the exchange he stressed the importance of communicating the universality of mathematical knowledge -that it’s not something unique to mathematicians, but is available to all.



I did not mean to imply that Ed Frankel was an advocate of philosophical platonism, and am grateful for his clarification. I presented the original piece as a reflection —philosophical reflection about some of the many interesting things in Frenkel’s LA Times piece. It’s perhaps a shortcoming of my training, and certainly an indication of my nominalist sympathies in the philosophy of mathematics that when the objectivity, persistence, timelessness, and universality of mathematical truths is on the agenda, my thinking turns to the evaluation of the entrenched philosophical views about these things.

For this reason I see the call to reevaluate mathematics education to be more open and in line with what mathematics really is and what mathematicians actually do as an opportunity to think philosophically about some of the important aspects of mathematical truth — aspects like universality, objectivity, and timelessness, and look at the entrenched philosophical views about these things in comparison to other philosophies that may be less other-worldly and mysterious.

And it is to that end that I wrote down some of my thoughts. And I hope to continue reflecting on mathematics and engaging with the work of educators and mathematicians like Ed Frenkel on these topics.

Axiomatizations of Arithmetic and the First-order/Second-order Divide

March 21, 2014

Catarina Novaes’ exciting paper on axiomatizations of arithmetic and the first-order/second-order divide, in four parts.


Philosophy and New Ways of Teaching Mathematics: Reflections on a piece by Frenkel, Part I

March 16, 2014

Update: see this for a clarification from Ed Frenkel regarding this post!

The LA Times recently published an article titled “How our 1,000-year-old math curriculum cheats America’s kids” by mathematician Ed Frenkel (1). The article is about changing the focus of early mathematics education to be more in line with what mathematics actually is and what mathematicians actually do rather than on the humdrum version of mathematics as memorization we all learned and or were terrified by when growing up.

Frenkel colorfully, and accurately in my view, compares the latter to learning to paint a wall in an art class, without ever seeing the works of great masters. He believes that this way of teaching math isn’t well suited to teaching the skill of abstraction, —bringing order to confusion by recognizing or establishing relations between things previously thought to be unrelated, that is important for the success of future generations. For Frenkel a broader view of mathematics is what’s needed for gifting this proficiency to students through “mathematical knowledge plus conceptual thinking times logical reasoning”. Part of the problem Frenkel sees is that “most of us never get to see the real mathematics because our current math curriculum is more than 1,000 years old” and focuses on numbers and solving equations instead of on concepts and ideas.

I take no issue with any of these claims and support the call to reform mathematics education by creatively introducing abstraction early on as an alternative to memorization for the sake of standardized tests. I’m writing this piece because of something philosophical that Frenkel says which Gives opportunity to discuss relevant philosophy of mathematics where I believe we may run the risk of tying new ways of teaching mathematics to future generations to highly speculative philosophy inspired by weighty, 2,000 year old metaphysics.

I’m thinking of this claim:

“We also need to convey to students that mathematical truths are objective, persistent and timeless. They are not subject to changing authority, fads or fashion. A mathematical statement is either true or false”.

The important philosophical ideas to discuss are:

  1. The claim about the objectivity, persistence and timelessness of mathematical truths.
  2. The characterization of mathematical truths as not subject to changing authority, fads, or fashion.
  3. The idea that the truth or falsity of mathematical statements stems from everyone agreeing to their truth or falsity —and how this squares away with ideas 1 and 2.

Focusing on 1, it’s generally accepted that most mathematicians behave as if mathematical truths and theorems are objective in some sense such that it makes sense to speak of their timelessness and persistence. The philosophical view commonly associated with actually believing these things this is a form of realism (the belief that the things, the objects, that mathematicians talk about are, in one sense or another, real things) called ‘platonism’. Platonism in this sense maintains that mathematical objects like numbers and sets exist independently of the human mind and are abstract, meaning that they don’t exist in time and space and are not causally related to things that do.

To be clear, this philosophy of mathematical platonism isn’t 2,000 years old, but it is inspired by that ancient philosophy of Plato’s about eternal, abstract, and immutable ideas (2). I think the modern term, ‘platonism’, as used in the philosophy of mathematics, was first used by the mathematician Paul Bernays who in the 1930’s used it to describe the tendency among mathematicians to treat mathematical things like sets and numbers like the abstract ideas of Plato (3). But long ago, Plato himself did rebuke mathematicians whose practice was not, in his view, sufficiently directed toward the eternal, abstract, and immutable ideas (4). My diagnostic comments below about mathematical platonism don’t pertain to any particular ideas of the historical Plato, but they’re hardly sympathetic to abstract metaphysics, something common to both mathematical platonism and Plato’s philosophy.

To me it seems strange to actually believe that through mathematical activity human beings somehow gain access to this disconnected realm of causally inert, immaterial objects existing beyond time and space. Mathematics, even in all it’s abstractness, seems to me to be a very natural and mundane human activity, like art, writing, or storytelling. I find it striking that through such earthly behavior an entire wholly alien and metaphysically disconnected universe is reached in such a way that in virtue of it, our mathematical theorems are true. I don’t mean to say that ordinary or basic human activities or things that are commonplace do not or cannot depend on or reveal complexity —but more that it seems out of place to bring in these in-principle disconnected and dubiously knowable things to account for common worldly behavior.

There’s a much discussed argument in defense of this type of realism. It’s the indispensability argument for ontological commitment. Ontological commitment is the idea that under a common understanding of how language works, sentences are committed to the existence of the things falling under the scope of the quantifiers if those sentences are to be true. The argument is simple: we must be ontologically committed to the things that are indispensable for our best scientific theories. Mathematical things like numbers and sets are indispensable for our best scientific theories. So we must be ontologically committed to mathematical objects.

It’s undoubtedly true that mathematics is part of the language of science but it still doesn’t follow that mathematical entities themselves are indispensable for science. Programs like the anti-realist fictionalism of Field (5) and the realist structuralism of Hellman (6) show, at least, that platonic objects aren’t indispensable to science and mathematics. I don’t think that in order to banish platonist philosophy in mathematics one should endorse that science somehow be awkwardly carried out without mathematics and that nominalists should drop everything and obtusely call for the translation of science into nominalist formalism instead of just normal mathematics —but the reality of nominalization undermines the indispensability claim on mathematical entities in science.

And the general claim about having to be ontologically committed to the existence of everything falling under the scope of the quantifiers we use when using mathematics in science seems to me also dubious. Maddy has non-trivially cast doubt on the type of support offered for the ontological commitment claim by showing that scientific theories aren’t treated as uniform by working scientists in the way that people who appeal to “our best scientific theories” would like to think (7). The point is that it’s a mixed bag when it comes to accepting the existence of mathematical and non-mathematical entities among scientists working on our bests theories and that scientists use mathematics to get results, without treating its applicability as a confirmation of its truth. So, working science doesn’t tell much about the truth of the mathematics, and much less so about the existence of mathematical entities. Thinking about it like this, it seems extravagant to insist on squeezing out the abstract metaphysics of platonism from the fairly worldly realm of science.

Still, if we’re just talking about mathematics and not so much about mathematics as used in science, then we still have the problem of mathematicians somehow having special access to the abstract, spatiotemporally disconnected entities in virtue of which, according to Platonism, mathematics is objective, persistent and timeless. Field famously streamlined an argument of Benacerraf addressing just this very point, concluding that if mathematical Platonism is true, then the reliability of mathematicians ascertaining mathematical truth can’t be explained (8). The reason it can’t be explained is that platonic entities are in principle spatiotemporally separate from the universe inhabited by mere mortals —apart from giving a mysterious account where mathematical knowledge is in a sense un-caused, or divined, it doesn’t look like there’s even an in-principle way of explaining the reliability of mathematicians’ ascertaining mathematical truth.

In any case, there are other more grounded ways to account for desirable qualities like, knowability, objectivity, persistence and timelessness of mathematical truth. For example, structuralism, the view that mathematics is about mathematical structures where the objects of mathematics are completely determined by their place in the structure, accounts for the knowledge we have of structural relations based on proofs from assumptions that provide for the types of structures under study. Further, such knowledge is objective because realist structuralism like Hellman’s maintains the determinateness of truth values that’s is so central to mathematics (9).

The relative timelessness and persistence of mathematical truths under such structuralism will just be a function of the timelessness of the logical possibility of the structures to be investigated —something that, along with mathematical possibility, is for the realist modal structuralist an irreducible primitive. A pretty good measure of timelessness and persistence, I think (10).

Frenkel takes no position on these issues, but I bring them up because I think it would be disappointing and run counter to teaching the skill of abstraction in a way so as to avoid, in Frenkel’s words, “misconceptions and prejudice” about mathematics if in answering to the objectivity, persistence and timelessness of mathematics we were to rely on deeply speculative philosophy. Frenkel relates how he used a Rubik’s cube to explain symmetry groups to fourth, fifth and sixth grade students. He also introduced them to “curved shapes (such as Riemann surfaces) and the three-dimensional sphere that give us glimpses into the fabric of our universe.” Relying on mathematical platonism as a philosophy runs counter to this because it communicates that ultimately even real mathematics that has to do with the real world, that kids are ready for and that can be practiced by people like you and me, is nevertheless mysterious, requiring special, unspecified access to a disconnected realm of causally inert, immaterial objects existing beyond time and space. And it makes little sense to do so, as there are operable, sophisticated and serious nominalist alternatives to platonism.

Platonism about mathematics is often also the go-to philosophy for justifying the idea that mathematical truths are not subject to changing authority, fads, or fashion, which is discussion point 2 above. It makes sense because mind-independent, spatiotemporally and causally disconnected things can’t really be affected by anything mere mortals can do, and aren’t subject to human authority, or passing fancy.

This is the idea that I’ll discuss in the next update.

2. For more about what the historical Plato had to say about these things see: (Kraut, 2013).
3. Bernays, 1935, p. 259.
4. Plato says in Republic, Book VII, 527b: “They [geometers] give ridiculous accounts of it [geometry], though they can’t help it, for they speak like practical men, and all their accounts refer to doing things. They talk of ‘squaring’, ‘applying’, ‘adding’, and the like, whereas the entire subject is pursued for the sake of knowledge…for the sake of knowing what always is” (Plato, 1997, p. 1143).
5. Field, 1980.
6. Hellman, 1989.
7. Maddy, 1992.
8. Field, 1989, p. 68.
9. Hellman, 1989, p. 44.
10. One may wonder at this point: Which logic? One of the virtues of modal structuralism is that because logical and mathematical possibility is irreducible, one is free to examine all kinds of structures, including constructive and paraconsistent ones.

Benacerraf, P. And H. Putnam, 1983, “Philosophy of Mathematics”, second edition, Cambridge: Cambridge University Press.

Bernays, P., 1935, “Platonism in mathematics,” in Benacerraf and Putnam, 1983, pp.258-271.

Cooper, John M. (ed.), 1997, Plato: Complete Works, Indianapolis: Hackett.

Field, H., 1980, “Science Without Numbers: A Defence of Nominalism”, Oxford: Blackwell.

—1989, “Realism, Mathematics, and Modality”, Oxford: Blackwell.

Hellman, G., 1989, “Mathematics without Numbers”, Oxford: Clarendon.

Kraut, R., 2013 “Plato”, The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), URL = <>.

Maddy, P., 1992, “Indispensability and Practice”, Journal of Philosophy, 89(6): 275–289.

Plato, 1997, “Republic”, in Cooper, John M. (ed.), 1997.

Weird Provability Distress Over Necessity

February 19, 2014

A philosophically inclined twitter follower mentioned me in a tweet yesterday expressing distress because using S5 modal logic you can derive necessary possibilities from possibilities in the special case where something like an unproven mathematical theorem is possible. If you take it that a mathematical claim, when a theorem, is necessary and it’s possible that a mathematical claim is a theorem, then by logic and the modal reduction property of S5 you can “prove” the theorem.

For example, say \phi is the theorem in question, \Box is necessity, and \diamond is possibility, \rightarrow is implication. Then you have \diamond(\Box\phi) \rightarrow \Box\diamond(\Box\phi), and by modal reduction you get \Box\phi, and therefore \phi.

Offering an S5 proof about the necessity of a desired theorem for any open problem in, say number theory, would be a miserable failure of mathematical reasoning for a variety of reasons, not the least of which is that it would reveal nothing about the mathematics in question; it would be an empty, deductive tautology in a logical system, unworthy of the name of mathematical truth.

The sneaky thing, in my opinion, about this reasoning is that it takes for granted that a mathematical claim, when true, is necessary, without a claim about its provability in a specific system of mathematics. Without further elucidation as to the meaning of a mathematical theorem’s possibility in a system, we end up operating on an alleged necessary truth in some vague, unspecified system of mathematics. What we really want to say is that if it’s possible that a claim is provable in a specific system of mathematics, like Peano Arithmetic, then so and so.

But this is just what it means to evaluate a mathematical system using provability logic: The \Box and \diamond operators can be fruitfully used in sophisticated logics to analyze the provability strength of mathematical systems like Peano Arithmetic. In these cases the \Box\phi means ‘\phi is provable in a system’ and \diamond\phi abbreviates \neg\Box\neg\phi, where \neg is negation. In these analyses the logic is a strengthened S4, which lacks the characteristic S5 axiom that is used in the above reasoning. The characteristic axioms for S4 provability logic are (1) \Box\phi \rightarrow \Box\Box\phi, which says that if \phi is provable in a system, then it’s provable that \phi is provable, and (2) \Box(\Box\phi \rightarrow \phi) \rightarrow \Box\phi, which is a famous theorem by Löb that says that if it’s provable that if \phi is provable then \phi is true, then \phi is provable.

But now we’re talking about what may or may not be proved in a given mathematical system, not about unproved necessities somehow cashing out as mathematical theorems, which I see as neither mathematics, nor an appropriate application of modal logic.

A Survey of Proofs on the Uncountability of the Unit Interval

January 23, 2014

If you have an infinitely long line of points, a segment that is 1 unit long has as many points as the whole line. Such a segment is not only infinite, but it’s infinite in a way that it cannot be counted with the counting numbers, 1, 2, 3, \dots and so on. The linked paper provides a comprehensive compendium of demonstrations of this last remarkable fact, ranging from the historical to algebraic and game theoretic proofs. Follow any one you like and convince yourself.

“The Uncountability of the Unit Interval”, by C. Knapp, C. Silva

Logicomix Available for Free Online Reading via Scribd

February 24, 2013

The graphic novel, “Logicomix: An Epic Search for Truth” is available for free online reading (pay for download) via Scribd.

It’s a fun story about the early search for mathematical foundations involving figures like Bertrand Russell, George Cantor, Ludwig Wittgenstein, Gottlob Frege, Kurt Gödel and Alan Turing.

And for those interested, here’s a review of Logicomix by noted philosopher of mathematics, Paolo Mancosu:

The Nonsense Math Effect

December 28, 2012

An article suggesting that non-math academics are impressed by equations even if the equations are nonsense:

Well Ordering Infinite Sets, the Axiom of Choice and the Continuum Hypothesis

December 17, 2012

We’re trying to understand the relationship between \aleph_{0}, and other infinite cardinal numbers. We’ve reviewed two ways (here and here) to generate infinite cardinals greater than \aleph_{0}. The first way is by considering the set of all subsets of positive integers, the power set \mathcal{P}(\mathbb{N}) of \mathbb{N} with cardinality 2^{\aleph_{0}}. The second way is by considering the set of countably infinite ordinals, \omega_{1} and its cardinality, \aleph_{1}.

We begin by introducing the axiom of choice. The axiom of choice (AC) is an axiom of set theory that says, informally, that for any collection of bins, each containing at least one element, it’s possible to make a selection of at least one element from each bin. It was first set forth by Zermelo in 1904 and was controversial for a time (early 20th century) due to its being highly non-constructive.

For example, AC is equivalent to the well ordering theorem (WOT) stating that every set can be well ordered, and it can be proven using AC/WOT that there are non-Lebesgue measurable sets of real numbers. Nevertheless, this result is consistent with the result that no such set of reals is definable! And use of the AC to achieve unintuitive mathematical results, like paradoxical decompositions in geometry in the spirit of the Banach-Tarski paradox, have also fueled skepticism and mistrust of the axiom.  But contemporary mathematicians make free use of the axiom and hardly mind that it was ever controversial.

Formally, the axiom of choice says the following:

For each family (X_{i})_{i \in I} of non-empty sets X_{i}, the product set \prod_{i \in I}{(X_{i})} is non-empty.

The elements of \prod_{i \in I}{(X_{i})} are actually “choice functions”
(x_{i})_{i \in I}, I \rightarrow \bigcup_{i \in I}{(x_{i})}, satisfying x(i) = x_{i} for each i \in I.

AC is important when thinking about infinite cardinals in part because of its equivalence to the WOT. Because WOT says that every set can be well ordered, it follows that each cardinal (any set really) can be associated with an ordinal number and you can count them via the ordinals. For instance, \aleph_{0} can be represented by \omega, \aleph_{1} by \omega_{1} and in general AC/WOT enables us to give the von Neumann definition of cardinal numbers where the cardinality of a set X is the least ordinal \alpha such that there is a bijection between X and \alpha.

AC and WOT do a lot of other work as well in terms of ordering infinities. We say that an aleph is the cardinal number of a well ordered infinite set. Because every set is well order able, all infinite cardinals are alephs. It was shown by Friedrich Hartogs in 1915 that trichotomy, which is a property of an order relation on a set A such that for any x, y \in A, x < y, x = y, or x > y, is equivalent to AC.

So with AC we can list, in (a total) order, the infinite cardinals and compare them. Thus we can set up the following infinite list of alephs, \aleph_{0}, \aleph_{1}, \aleph_{2}, \cdots, \aleph_{\alpha}, \cdots. We know that \aleph_{1} is greater than and distinct from \aleph_{0}. The cardinal \aleph_{1} is the cardinality of the ordinal \omega_{1} which is larger than all countable ordinals, so \aleph_{1} is distinguished from \aleph_{0}. What about 2^{\aleph_{0}}?

It’s unclear where 2^{\aleph_{0}} fits in in the list \aleph_{0}, \aleph_{1}, \aleph_{2}, \cdots, \aleph_{\alpha}, \cdots. The infinite cardinal 2^{\aleph_{0}} is the cardinality of the set of subsets of \mathbb{N}, but it’s also the cardinality of the set, \mathbb{R}, of real numbers or the set of points on a line, known as the continuum. (This equality is provable using the Cantor-Schröder-Bernstein theorem and follows from the proof of the uncountability of \mathbb{R}.)

George Cantor, the inventor of set theory, conjectured in 1878 that there is no set whose cardinality is strictly between the cardinality of the integers (\aleph_{0}) and the cardinality of the continuum (2^{\aleph_{0}}). Since we’re assuming AC, that means that 2^{\aleph_{0}} = \aleph_{1}.

This conjecture is famously known as the Continuum Hypothesis (CH) and was the first of 23 problems in David Hilbert’s famous 1900 list of open problems in mathematics. The problem of whether or not CH is true remains open to this day although Kurt Gödel (in 1940) and Paul Cohen (in 1963) showed that CH is independent of the axioms of Zermelo-Fraenkel set theory, if these axioms are consistent.

There is much of philosophical interest surrounding the mathematics of the continuum hypothesis and I hope to be able to turn my attention to those topics in the future.

Priest, Beall and Armour-Garb’s “The Law of Non-Contradiction” Available Online via Scribd

November 5, 2012

Priest, Beall and Armour-Garb’s “The Law of Non-Contradiction”, (link below) is a great collection of essays on the philosophy and logic of dialetheism, the belief that that there are sentences A, such that both A and its negation, ¬A are true. Non-classical, paraconsistent logics may be necessary for formalizing and understanding physical and social systems.

Paul Cohen Reflects on the Nature of Mathematics

November 5, 2012

Reflections on the nature of mathematics from Paul Cohen’s “Comments on the foundations of set theory” in Scott and Jech, Axiomatic Set Theory, Vol.1, p. 15.