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> 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.

• tiahura 1y ago
  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.
• myrmidon 1y ago
  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.
  • TimPC 1y ago
    In general you aren't testing as an empiricist though, you are looking for a rational argument to prove or disprove something.
    • Tazerenix 1y ago
      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
      • 2snakes 1y ago
        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/10.3389/fpsyg.2022.890362/full https://www.frontiersin.org/journals/psychology/articles/10....
  • pdpi 1y ago
    > 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.
• The_suffocated 1y ago
  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.
• AlexandrB 1y ago
  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_theorems https://en.wikipedia.org/wiki/Gödel%27s_incompleteness_theor...
• _9ptr 1y ago
  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.
• weinzierl 1y ago
  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.
  • lo_zamoyski 1y ago
    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.)
    • weinzierl 1y ago
      Well, we are talking about pure mathematics and there is not much Popperian scientific method in it.
      • Warwolt 1y ago
        Who cares? That's just semantics. If we define science as the systematic search for truths, then mathematics and logic are the paradigmic sciences. If we define it as only empirical search for truth then perhaps that excludes mathematics, but it's an entirely unintersting point, since it says nothing.
• lo_zamoyski 1y ago
  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).
• ubj 1y ago
  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.
• abdullahkhalids 1y ago
  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.
  • goatlover 1y ago
    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.
    • naasking 1y ago
      > Plenty of mathematical proofs have been proven true with 100% certainty Solipsists would like to have a word with you...
      • griffzhowl 1y ago
        Tell them to mind their own business ;
    • deleted 1y ago
      [deleted]
  • drdeca 1y ago
    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”.
  • griffzhowl 1y ago
    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
• Aardwolf 1y ago
  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
  • griffzhowl 1y ago
    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.
• nitwit005 1y ago
  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.
  • goatlover 1y ago
    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.
    • nitwit005 1y ago
      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.
  • griffzhowl 1y ago
    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
    • Jensson 1y ago
      > 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.
      • griffzhowl 1y ago
• mcmoor 1y ago
  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.