# What does “most of mathematics” mean? [closed]

After reading the question Is most of mathematics not dealing with sets? I noticed that most posters of answer or comments seemed to be comfortable with the concept of "most of mathematics".

I'm not trying to ask a stickler here, I'm just curious if there is some kind of consensus how the quantification of mathematics might be done. Is the fraction's denominator only known mathematics in 2017, or all of Mathematics, from a potential viewpoint that math exists whether we realize it or not.

For example, the highly up voted, accepted answer begins:

"It is very well-known that most of mathematics..."

which at least suggests some level of consensus. In this case, how does the consensus agree that math is quantified?

I hesitated to ask this question at first, wondering if answers would be potentially too opinion based. So I'd like to stick to answers that address the existence of some kind of consensus how the quantification of mathematics might be done.

For example. does the question "Does at least half of mathematics involve real numbers?" even make sense? If so, could it actually mean something substantially different to each individual who believes it makes sense? Or in fact is there at least some kind of consensus.

• I'm looking forward to Noah Schweber's answer to this question, which will invariably contain a phrase that shall be singled out for a new question ... :) – pjs36 Feb 22 '17 at 14:01
• "Is the fraction's denominator..." Which fraction? – Lee Mosher Feb 22 '17 at 14:21
• I think that you have to read it as : Open some math textbook : calculus, analysis, algebra; for sure in the first or initial few pages you will find a statement like: "Let $S$ a set of (numbers, function, etc.)". But in most cases you will not find a discussion about "what set are ?" "what are the relevant properties of sets", etc. At most, you would find in the textbook an Appendix dedicated to Elements of Logic and S.T.. – Mauro ALLEGRANZA Feb 22 '17 at 14:28
• Put in otehr terms, for sure "set language" is used as universal language for mathematics, but most of "working math" does not needs set theory as foudations (in the technical use of the term). This, I think, is the gist of Noah's initial statement : "It is very well-known that most of mathematics does not care about foundational matters..." – Mauro ALLEGRANZA Feb 22 '17 at 14:31
• "most of mathematics" means the half of whole possible theorems + 1 – Dac0 Feb 22 '17 at 17:53

It is clear that "most of mathematics" means "the half of whole possible theorems + 1"...

Jokes aside I think that when someone is saying "most of mathematics" is in fact referring to mathematics as an activity and not to mathematics as a subject.

So when someone is saying that "most of mathematics doesn't care about foundational mathematics..." he's just saying that most of the mathematical activity that is ongoing nowday by professional mathematicians is not focused on foundational subtleties and even doesn't care about it. So you can safely be a working mathematician and not knowing ZFC.

So I think you question is just a misunderstanding between mathematics as a subject and mathematics as an activity.

• Your explanation is very helpful, and I think you've cut to the chase nicely. Thanks! – uhoh Feb 22 '17 at 18:22
• Technically, half of all possible theorems + 1/2 is already most of mathematics. – Federico Poloni Feb 22 '17 at 19:16
• :D :D :D you're right – Dac0 Feb 22 '17 at 19:29
• @uhoh: While mathematical activity might be what some people mean when they say "most mathematics", this is very much weaker than what some logicians mean (like Friedman and Simpson), as I've explained in my answer. Dac0's version could still hold even if most mathematical activity logically relies crucially on the ZFC axioms. But the actual fact is that, regardless of what professional mathematicians focus on, their theorems hardly need the full strength of ZFC. So differentiating between the subject and the activity is a red herring, because even the subject mostly does not need ZFC. – user21820 Feb 23 '17 at 3:35
• @uhoh: In my comment I did not at all say anything about whether mathematics exists independent of people, so I don't know why you brought up that notion. My point is simply that this post is misleading concerning the quoted phrase. It misrepresents what many logicians actually intend to convey by "most mathematics". I'm not sure if you know about reverse mathematics, but you should take a look at the Wikipedia page that I linked from my answer to the other question to see examples of well-known theorems that are provable in very weak systems, to see why they don't just refer to "activity". – user21820 Feb 23 '17 at 7:07

I can't speak for other answers to the linked question, but for my answer I was referring to most of mathematics as is currently accepted as of today. To make this precise there are two points that we need to clarify. In my answer, I linked to a precise definition for interpretability, so that it doesn't matter what specific language we are using. This is important because one could otherwise argue that mathematicians who do not write in the language of set theory (like many in history) are not actually dealing with sets. The second important way one can make the idea of currently accepted mathematics precise is to specify precisely a collection of theorems. For example we have Harvey Friedman's Grand Conjecture:

Every theorem published in the Annals of Mathematics whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of Peano Arithmetic based on the usual quantifier-free axioms for $0, 1, +, ×, exp$, together with the scheme of induction for all formulas in the language all of whose quantifiers are bounded.

In this post he further adds: "sometimes with qualifications that the article not be written by people referring to themselves as logicians". I think the general consensus is that there are likely to be only few counter-examples that are not contrived. Note that EFA is far weaker than ACA (which I was referring to in my linked answer). Also, the Wikipedia article mentions some natural mathematical problems where the answer cannot be proven in EFA, such as the optimal asymptotic complexity of a disjoint-set data structure, in this case because it involves the Ackermann function, which grows faster than any provably total function in EFA.

The reason why I single out ACA is because there is a natural correspondence between arithmetical sets (namely sets of natural numbers definable by an arithmetical formula over PA) and oracles for finite iterations of the halting problem. It may be a surprising that this is all we need to encode a vast portion of ordinary mathematics, which according to Stephen Simpson refers to "mathematics which is prior to or independent of the introduction of abstract set-theoretic concepts, [including] such branches as geometry, number theory, calculus, differential equations, real and complex analysis, countable algebra, the topology of complete separable metric spaces, mathematical logic, and computability theory". In contrast the Grand Conjecture is restricted to arithmetical sentences, because real numbers cannot be encoded in EFA.

To have a consensus about the numerical fraction meant by "most of mathematics" you'd have to have a reasonable number of mathematicians who are interested enough in that fraction to express an opinion. I doubt that there are many such mathematicians.

Noah Schweber's excellent upvoted accepted answer you refer to explains why.

If you did really want numbers, you might use for the denominator the total number of mathematical journal articles or pages in a recent year, with the number devoted to foundations for the numerator.

Here are the set theory tag statistics for this site, divided by the statistics for the site as a whole:

Set theory tag: 447 followers, 4.2k questions
Site total: ~13,000 users, 738,074 questions

So about 3.4% of the users, 0.57% of questions.

The percentages may differ so much because many questions are from students, not yet "mathematicians", who ask about things that come up in their studies, while users who answer often and take the trouble to follow a tag are more likely professionals.

• For papers as a sort of "pulse", you might use the (most recent) 2010 Mathematics Subject Classification. Although there's such a variety of journals, I would be surprised if you could get much better than a rough estimate of, say, articles classified under Measure Theory and Integration (to say nothing of the total number of articles published in a given timeframe). – pjs36 Feb 22 '17 at 14:11
• @uhoh See my edit. Hope it clarifies my meaning. – Ethan Bolker Feb 22 '17 at 14:28
• Very nicely, thanks! – uhoh Feb 22 '17 at 14:34