> Pure mathematics is regarded as an abstract science, which it is by definition.

I'd argue that, by definition, mathemtatics is not, and cannot be, a science. Mathematics deals with provable truths, science cannot prove truth and must deal falsifiability instead.

You could turn the argument around and say that math must be a science because it builds on falsifiable hypotheses and makes testable predictions.

In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit; can be answered both yes and no using reasonable definitions.

> In the end arguing about whether mathematics is a science or not makes no more sense than bickering about tomates being fruit

That's the thing, though — It does make sense, and it's an important distinction. There is a reason why "mathematical certainty" is an idiom — we collectively understand that maths is in the business of irrefutable truths. I find that a large part of science skepticism comes from the fundamental misunderstanding that science is, like maths, in the business of irrefutable truths, when it is actually in the business of temporarily holding things as true until they're proven false. Because of this misunderstanding, skeptics assume that science being proven wrong is a deathblow to science itself instead of being an integral part of the process.

In general you aren't testing as an empiricist though, you are looking for a rational argument to prove or disprove something.

The practical experience of doing mathematics is actually quite close to a natural science, even if the subject is technically a "formal science* according to the conventional meanings of the terms.

Mathematicians actually do the same thing as scientists: hypothesis building by extensive investigation of examples. Looking for examples which catch the boundary of established knowledge and try to break existing assumptions, etc. The difference comes after that in the nature of the concluding argument. A scientist performs experiments to validate or refute the hypothesis, establishing scientific proof (a kind of conditional or statistical truth required only to hold up to certain conditions, those upon which the claim was tested). A mathematician finds and writes a proof or creates a counter example.

The failure of logical positivism and the rise of Popperian philosophy is obviously correct that we can't approach that end process in the natural sciences the way we do for maths, but the practical distinction between the subjects is not so clear.

This is all without mention the much tighter coupling between the two modes of investigation at the boundary between maths and science in subjects like theoretical physics. There the line blurs almost completely and a major tool used by genuine physicists is literally purusiing mathematical consistency in their theories. This has been used to tremendous success (GR, Yang-Mills, the weak force) and with some difficulties (string theory).

————

Einstein understood all this:

> If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? Nay, more, has this right way any existence outside our illusions? Can we hope to be guided safely by experience at all when there exist theories (such as classical mechanics) which to a large extent do justice to experience, without getting to the root of the matter? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. Our experience hitherto justifies us in believing that nature is the realisation of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. Experience may suggest the appropriate mathematical concepts, but they most certainly cannot be deduced from it. Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed. - Albert Einstein

An alternative to abstraction is to use iconic forms and boundary math (containerization and void-based reasoning). See Laws of Form and William Bricken's books recently. Using a unary operator instead of binary (Boolean) does indeed seem simpler, in keeping with Nature. Introduction: https://www.frontiersin.org/journals/psychology/articles/10....

Mathematical "truth" all depends on what axioms you start with. So, in a sense, it doesn't prove "truth" either - just systemic consistency[1] given those starting axioms. Science at least grapples with observable phenomena in the universe.

[1] And even this has limits: https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...

Mathematical proofs are checked by noisy finite computational machines (humans). Even computer proofs' inputs-outputs are interpreted by humans. Your uncertainty in a theorem is lower bounded by the inherent error rate of human brains.

I agree we can't be absolutely certain of anything (maybe none of our memories are real and we just popped into existence etc.)

But we can be more sure of the deductive validity of a proof than we can be of any of the claims you make in these sentences, so I don't think they can serve to establish any doubt. If we're wrong about deductive logic, then we can only be more wrong about any empirical claims, which rely on deductive logic plus empirical observations

Plenty of mathematical proofs have been proven true with 100% certainty. Complicated proofs that involve a lot of steps and checking can have errors. They can also be proven true if exhaustively checked.

> Plenty of mathematical proofs have been proven true with 100% certainty

Solipsists would like to have a word with you...

Tell them to mind their own business ;

This may be, but not, I think, in a way that is particularly worth modeling?

When we try to model something probabilistically, it is usually not a great idea to model the probability that we made an error in our probability calculations as part of our calculations of the probability.

Ultimately, we must act. It does no good to suppose that “perhaps all of our beliefs are incoherent and we are utterly incapable of reason”.

You're saying maybe people have mistakenly accepted incorrect proofs now and again, so some theorems that people think are proven are unproven. I agree that this seems very likely.

In practice when proofs of research mathematics are checked, they go out to like 4 grad students. This isn't a very glamorous job for those grad students. If they agree then it's considered correct...

But note this is just the bleeding edge stuff. The basic stuff is checked and reproven by every math undergrad that learns math. Literally millions of people have checked all the proofs. As long as something is taught in university somewhere, all the people who are learning it (well, all the ones who do it well) are proving / checking the theory.

Anyway, when the scientific community accepts a bad proof what effectively happens is that we've just added an extra axiom.

Like when you deliberately add new axioms, there are 3 cases

- Axiom is redundant: it can be proven from the other axioms. (this is ... relatively fine? we tricked ourselves into believing something that is true is true, the reason is just bad.)

This can get discovered when people try to adapt the bad proof to prove other things and fail.

Also people find and publish and "more interesting", "different" proofs for old theorems all the time. Now you have redundancy.

- Axiom contradicts other axioms: We can now prove p and not p.

I wonder if this has ever happened? I.e. people proving contradictions, leading them to discover that a generally accepted theorem's proof is incorrect. It must have happened a few times in history, no?

o/c maybe the reason this hasn't happened is that the whole logical foundation of mathematics is new, dating back to the hilbert program (1920s).

There are well known instances of "proofs" being overturned before that, but they're not strictly logically proofs in the hilbert-program sense, just arguments. (Of course they contain most of the work and ideas that would go into a correct proof, and if you understand them you can do a modern proof)

e.g. https://mathoverflow.net/a/35558

Cauchys proof that, if a sequence of continuous functions converges [pointwise] to a function, the limit function is also continuous (cauchys proof only holds for uniform convergence, not pointwise convergence - but people didnt really know the difference at the time)

- Axiom is independent of other axioms: You can't prove or disprove the theorem.

English doesn't have a "I'm just hypothesizing all of this" voice, if it did exist this post should be in it. I didn't do enough research to answer your question. Some of the above may be wrong, e.g. the part about the 4 grad students. One should probably look for historical examples.

Mathematics is a science of formal systems. Proofs are its experiments, axioms its assumptions. Both math and science test consistency—one internally, the other against nature. Different methods, same spirit of systematic inquiry.

It's not an empirical science, but it is a science, where "science" means any systematic body of knowledge of an aspect of a thing and its causes under a certain method. (In that sense, most of what are considered scientific fields are families of sciences.) Mathematics is what you'd call a formal science with formal structure and quantity as its object of study and deductive inference and analysis as its primary methods (the cause of greatest interest is the formal cause).

A proof is just an argument that something is true. Ideally, you've made an extremely strong argument, but it's still a human making a claim something is true. Plenty of published proofs have been shown to be false.

Math is scientific in the sense that you've proposed a hypothesis, and others can test it.

Difference is mathematical arguments can be shown to be provably true when exhaustively checked (which is straight forward with simpler proofs). Something you don't get with the empirical sciences.

Also the empirical part means natural phenomena needs to be involved. Math can be purely abstract.

You're making a strong argument if you believe you checked every possibility, but it's still just an argument.

If you want to escape human fallibility, I'm afraid you're going to need divine intervention. Works checked as carefully as possible still seem to frequently feature corrections.

The difference is that in mathematics you only have to check the argument. In the empirical sciences you have to both check the argument and also test the conclusion against observations

> In the empirical sciences you have to both check the argument and also test the conclusion against observations

That isn't true, you just test new axioms but most stuff we do in empirical sciences don't require new axioms.

The only difference between material sciences and math is that in math you don't test axioms while in empirical sciences you do.

Empirical science uses both deductive logic to make predictions, and observations to check those predictions. I'm not saying that's all it involves. Not sure which part of that you disagree with

And a lot of what goes on in foundations of mathematics could be described as "testing the axioms", i.e. identifying which theorems require which axioms, what are the consequences of removing, adding, or modifying axioms, etc.

Somewhat tangential to the discussion: I have once read that Richard Feynman was opposed to the idea (originally due to Karl Popper) that falsifiability is central to physics, but I haven't read any explanation.

I'm not sure if it deals only with provable truths? It even deals with the concept of unprovability itself, if the incompleteness theorem is considered part of mathematics

Yes, but Godel proved the incompleteness theorem, by ingeniously finding ways to prove things about unprovability.

The incompleteness theorem doesn't say that there are statements which are unprovable in any absolute sense. What it says is that given a formal system, there will always be statements which that particular formal system can't prove. But in fact as part of the proof, Godel proves this statement, just not by deriving it in the formal system in question (obviously, since that's what he's proving is impossible).

The way this is done is by using a "metalanguage" to talk about the formal theory in question. In this case it's a kind of ambient set theory. Of course, the proof also implies that if this ambient metalanguage is formalized then there will be sentences which it can't prove either, but these in general will be different sentences for each formalized theory.

It isn't, that's why it's in own section in STEM, and rightfully so. It's a higher tool that without it, science would come to a screeching halt.

Science involves both deductive and inductive reasoning. I would in turn argue that mathematics is a science that focuses heavily (but not entirely) on deductive reasoning.

He probably means science in a wider sense as opposed to the anglo-american narrower sense where science is just physics, chemistry, biology and similar topics.

Pure mathematics is just symbol pushing and can never be science. It is lot of fun though and as it turned out occasionally pretty useful for science.

It is absolutely a science, a formal science. What it isn't is an empirical science.

The "symbol pushing" is a methodological tool, and a very useful one that opened up the possibility of new expansive fields of mathematics.

(Of course, it is important to always distinguish between properties of the abstraction or the tool from the object of study.)

Well, we are talking about pure mathematics and there is not much Popperian scientific method in it.

I agree in general but

> Euclid's Elements is 2300 years old and is presented in a completely abstract way.

depends on what you mean by completely abstract. Euclid relies in a logically essential way on the diagrams. Even the first theorem doesn't follow from the postulates as explicitly stated, but relies on the diagram for us to conclude that two circles sharing a radius intersect.

This is a thought-provoking paper on the issue by Viktor Blasjo, Operationalism: An Interpretation of the Philosophy of Ancient Greek Geometry https://link.springer.com/article/10.1007/s10699-021-09791-4

which was recently the subject of a guest video on 3blue1brown https://www.youtube.com/watch?v=M-MgQC6z3VU

Not only is intuition important (or the entire point; anyone with some basic training or even a computer can follow rules to do formal symbol manipulation. It's the intuition for what symbol manipulation to do when that's interesting), but it is literally discussed in a helpful, nonjudgmental way on Math Stack Exchange. e.g.

https://math.stackexchange.com/questions/31859/what-concept-...

Other great sources for quick intuition checks are Wikipedia and now LLMs, but mainly through putting in the work to discover the nuances that exist or learning related topics to develop that wider context for yourself.

> The next paragraph about how mathematics was closely coupled to reality for most of history and only recently with our understanding of infinite sets became too abstract is not really at all accurate of the history of mathematics. Euclid's Elements is 2300 years old and is presented in a completely abstract way.

I may be off-base as an outsider to mathematics, but Euclid’s Elements, per my understanding, is very much grounded in the physical reality of the shapes and relationships he describes, if you were to physically construct them.

Quite the opposite, Plato, several hundred years before Euclid was already talking about geometry as abstract, and indeed the world of ideas and mathematics as being _more real_ than the physical world, and Euclid is very much in that tradition.

I am going to quote from the _very beginning_ of the elements:

Definition 1. A point is that which has no part. Definition 2. A line is breadthless length.

Both of these two definitions are impossible to construct physically right off the bat.

All of the physically realized constructions of shapes were considered to basically be shadows of an idealized form of them.

Another point to keep in mind is that a lot of mathematics that's not considered abstract _now_ was definitely considered "hopelessly" abstract at the time of its conception.

The complex number system started being explored by the greeks long before any notion of the value of complex spaces existed, and could be mapped to something in reality.

I don't think we can say the Greeks were exploring complex numbers. There's something about Diophantus finding a way to combine two right-angled triangles to produce a third triangle whose hypotenuse is the product of the hypotenuses of the first two triangles. He finds an identity that's equivalent to complex multiplication, but this is because complex multiplication has a straighforward geometric interpretation in the plane that corresponds to this way of combining triangles.

There's a nice (brief) discussion in section 20.2 of Stillwell's Mathematics and its History

Hell, 0 used to be considered too abstract!

Plato was only about a generation before Euclid. Their lives might have even overlapped, or nearly so: Plato died in 347BC and Euclid's dates aren't known but the Elements is generally dated ~300BC

The only things that are weird in math are things that would not be expected after understanding the definitions. A lot of the early hurdles in mathematics are just learning and gaining comfort with the fact that the object under scrutiny is nothing more than what it's defined to be.