23

I've reading many entries about whether Math uses the scientific method and the dominant opinions seems to be "no", e.g. from "Is Mathematics a science?" and other websites.

  • James Moosh, PhD in Pure Maths

    1) The use of the scientific method of theorising based on empirical data gained from rigorous and repeatable experimentation. 2) Using this method to create theories to explain natural phenomena.

    Mathematics doesn't do either of those things.

  • Alexander Farrugia, PhD in Mathematics

    In short, because it doesn’t use the scientific method.

    The scientific method is empirical: conclusions are made through observations and experiments. These conclusions may be (and often are) the best fit for the observations or experiments, and they may remain unchallenged for years, even centuries. However, they may not be true, per se: more observations and experiments may result in different conclusions which fit them better.

    Mathematics is deductive. Conclusions are made through a logical argument, by applying axioms and inference rules. The conclusions are true, as long as the person reading the proof agrees with the inference rules (and with the axioms).

However, you also find things like Using the Scientific Method to Engage Mathematical Modeling: An Investigation of pi

also,

As expressed by Paul Halmos: "Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork".

Experimental Mathematics

Also, there is this example, "The simplest mathematical comprobation". It's in Spanish, but it is understandable through the picture. When I have to check with my eyes the angles add up to 180 degrees, how isnt that an empirical comprobation?

The simplest mathematical comprobation

So, does it use it, does it not use it, or does it use it only in certain fields of Mathematics?

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  • 1
    Scientific Method is quite a broad concept, but observation and experiment are essential to it. Mar 13 at 14:18
  • 5
    I think observation and experiment actually ARE part of math, and central to many mathematical discoveries. @MauroALLEGRANZA
    – TKoL
    Mar 13 at 15:11
  • 1
    This is an interesting question which is not as easy at it may look. For example, the two very quick answers which have been posted are based on the axiomatic and inductive construction of mathematics, failing to observe that the mathematical practice (that is, maths as practised by mathematicians) is widely different from its logical formalization. Nobody works inductively from the axioms of ZFC but rather with some kind of trial and error which, as TKoL observes, may be akin to experimenting. This an idea that Cavaillès developed in "La pensée mathématique"—I'll try to write an answer.
    – Johan
    Mar 13 at 15:20
  • 1
    At the same time, "mathematics is the queen of science (C.F.Gauss)"; thus scientific method is not all in science. Mar 13 at 15:28
  • 5
    Science is an empirical discipline (arguably, the empirical discipline), whereas mathematics is a logical discipline. "Empirical" is NOT "logical", they are contrasting processes. So NO, theoretical mathematics does not employ the scientific method because mathematics is not a Science (it is technically a type of philosophy). Applied mathematics is a different matter, because it straddles the divide between the two. Mar 14 at 18:07

18 Answers 18

27

I'm going to go against common opinion here and say that, while mathematical truths may be considered distinct from scientific truths (and I'm not disagreeing with that), the process of discovering mathematical truths may follow science-like processes.

First of all, obviously peer review is part of academic mathematics. It has that in common with the scientific method.

Secondly, and this isn't true for ALL mathematical truths, but some mathematical truths are in part found by a process of trial and error, and measurement and observation. Think about pi, for example - early estimates for pi were made by measuring. Think about a^2 + b^2 = c^2 - the intuition to come up with this idea was probably inspired by many observations, and crucially, the result was then tested. Someone could scientifically falsify if a^2 + b^2 = c^2, by constructing a real right triangle for which it wasn't true.

Many, but not all, mathematical discoveries are intuited from many observations, and are testable and in principle falsifiable by other observations.

I think even derivatives and integrals from calculus were discovered and tested through science-like methods.

I'm not saying all of mathematics is science, but I am saying that a significant portion of mathematics follows science-like processes to generate and verify ideas.

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    I made a comment similar to yours in one of the answers. For example, if I recall correctly, (something done over 25 years ago) when asked to demonstrate "the determinant of a triangular matrix is the product of the entries on the main diagonal", we did trial and error of replacements and things like that to reach the final equation from the more general equation for all matrix
    – Pablo
    Mar 13 at 20:35
  • 5
    I found something interesting en.wikipedia.org/wiki/Experimental_mathematics . "As expressed by Paul Halmos: "Mathematics is not a deductive science—that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork". Paul Richard Halmos was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis
    – Pablo
    Mar 14 at 14:46
  • 3
    Zeilberger wrote a fairly strong statement in favor of experimental mathematics as an alternative to pure mathematics in AMS Notices about ten years ago, there's a good summary of it with a link to the full text of his essay here: experimentalmath.info/blog/2013/12/…
    – Idran
    Mar 14 at 20:35
  • 5
    As someone who has written mathematical proofs, I can attest to the fact that even pure mathematicians often experiment with different logical arguments, some of which are failures, before the arrive at a correct proof. You could safely say that they study, form a hypothesis, test it (by trying things), analyze their results, and report their conclusions. Many comments say that the proofs in mathematics are the equivalent to the scientific method, this is not the case, the proof is only the conclusion, not the whole thing.
    – Mathaddict
    Mar 14 at 22:07
  • 3
    @Pablo My point was that Gauss was not proposing a mathematical experiment at all. If you were to construct a triangle for which pythagoras did not hold, then it would not be a plane triangle. You can't disprove the theorem (because it is a theorem of Euclidean geometry). That's what being a theorem is all about. Gauss was interested in knowing whether space was curved and therefore if you triangulated three separated point, would they indicate it. He was applying maths not testing it. Mar 15 at 6:02
12

The relationship between mathematics and the scientific method is a complex and debated topic. Here's my perspective on it:

  1. Pure mathematics, which deals with abstract concepts, axioms, and logical deductions, does not typically employ the scientific method in its purest form. The scientific method is primarily an empirical approach based on observations, formulating hypotheses, conducting experiments, and drawing conclusions from the data.

  2. However, certain branches of mathematics, particularly those related to mathematical modeling, applied mathematics, and statistics, do incorporate aspects of the scientific method.

  3. In fields like mathematical physics, mathematical biology, or operations research, mathematicians often collaborate with scientists to develop mathematical models that describe real-world phenomena. In these cases, the scientific method is used to gather data, formulate hypotheses (mathematical models), test the models against experimental or observational data, and refine or revise the models as necessary.

  4. Even in pure mathematics, there is an element of exploration, conjecture, and proof that resembles aspects of the scientific method. Mathematicians may observe patterns, make conjectures (hypotheses), and then attempt to prove or disprove them through logical reasoning and deduction (analogous to experiments and analysis in science).

  5. Additionally, the process of peer review and validation in mathematics, where published proofs and results are scrutinized by the mathematical community, bears some resemblance to the replication and verification process in science.

In summary, while pure mathematics does not strictly follow the scientific method, there are certain areas of applied mathematics and mathematical modeling where the scientific method is employed to varying degrees. Furthermore, the general approach of observation, conjecture, and rigorous proof in mathematics shares some philosophical similarities with the scientific method, even if the specific methodologies differ.

11

In my view, there is a false dichotomy muddying this issue. No, mathematics does not use the scientific method in the same sense as the natural sciences. The results of experiments do not count as proof in mathematics; the only reasoning that is accepted as rigorously establishing results is deductive.

On the other hand, comments like Halmos's seem to contradict this, but what I think Halmos is really referring to is the creative process by which we discover a proof. To be honest, I wish people would be more careful how they express such views, which are generally directed to people without experience or advanced knowledge of math. Granted, we want people to know that research in math is not a mindless, procedural activity. On the other hand, I see no reason to be loose and cavalier about it.

I certainly don't think that even Halmos's comment is supporting a view that mathematics uses the scientific method. The discovery process is very much like an artistic creative process. Many mathematicians engage in artistic activities (like writing music, painting, etc.) as hobbies. The process of writing music, for instance, is closely analogous to math research. You have a body of knowledge coming from established practice (standard harmony, counterpoint, etc.). To this, you add some fiddling around, trial and error, etc. The end result stands on its own as a piece of music. We don't judge its value or validity according to the scientific method or even anything about the creative process.

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    Good stuff +1. I was quite amused at the answers so far persist in entangling of the false dichotomy you allude to. To which I would say: (1) Yeah mathematicians are humans — surprise! surprise!! (2) Still to say that mathematicians follow the same creative process as empirical scientists is somewhat tantamount to saying: Jazz players are creative, footballers are creative. So jazz players are footballers! I had an answer here which the mods have chosen to delete. Anyways it was a further development of the basic core that is still here...
    – Rushi
    Mar 14 at 18:49
  • ... That content (Geremias ref to a Christian-Aristotelian-Platonist) actually shows the way towards disentangling
    – Rushi
    Mar 14 at 18:54
5

Yes, but not in the way you might think.

The primary activity of mathematics is not being given a candidate theorem and asked to prove it - indeed that doesn't really involve the scientific method. It's discovery of the useful definitions, axioms, formalizations, relationships, etc. that lead to results that are interesting, beautiful, or applicable to important domains of human activity. Large parts of the history of mathematics have been about how to make a particular concept rigorous. And the way this problem is solved is largely the scientific method: mathematicians put forth (either published, or in their own minds) some idea, and it's tested for what comes of it. It may be a dead end, or may have no immediate clear value, or may turn out to solve problems people have been banging their heads over for a long time, or may just offer a completely different approach to something mathematicians already know how to do in a different way, or may connect seemingly-unrelated areas of mathematics.

Some examples that come to mind:

  • Epsilon-delta / making calculus rigorous
  • Lebesgue integral
  • Complex numbers
  • Axiomatic set theory
  • Nonstandard analysis
  • Complexity classes
  • Galois theory
5

The authors essentially claim that mathematics is not inductive; even if that were true (but it clearly isn't), it would still use the other half of the scientific method, deduction. Great scientific progress has been made by deduction as well, prominently special and general relativity; Einstein took what was known at the time and logically deduced the consequences, regardless of how absurd they seemed to the contemporary physicist. Experiments needed decades to actually verify many predictions made by general relativity. Nobody in their right mind would claim that General Relativity is not science just because it was mostly deduced, and the experiment came after the theory.

But it is also plain wrong to say that mathematics does not use "empirical data", does no experimentation and thus does not use induction. We just need to expand or generalize the term "empiric": True, pure mathematics deals with "objects" or concepts which are not present in the physical world and thus cannot be perceived. But these mental entities can be manipulated in our minds just like (or even better!) than objects in the physical world. A Gedankenexperiment can be observed and yield as many insights as any other. We have mental "experiences" and thus have mental "empirical" data, if we widen the term a bit beyond the definition your rather dull and dogmatic authors used. Additionally, nature "models" mathematics so that, in fact, physical observation and measurement can, in fact, lead to mathematical theories.

"Observations" made of real-world or imagined mathematical objects (all triangles drawn in a circle with the long side crossing the center and all corners lying on the perimeter are rectangular; there doesn't seem to be a highest prime, or even a highest twin prime etc.) lead to hypotheses which are then proven, very much like in other sciences. That is textbook induction. And of course one performs "experiments" by trying to concoct counter-examples.

The distinction is artificial and does not hold up to scrutiny.

4

Of course mathematics is a science. There isn't the slightest doubt about that.

But it's different from other sciences, in which there is no such thing as proof. Other sciences have to examine solid objects (or elementary particles) and use statistics to draw conclusions. By contrast, mathematics reaches its conclusions via proof.

But mathematics is nevertheless a rigorous study of a specific area, based on facts and logic, and almost completely independent of opinion. That's what makes it a science.

(Whether it uses "the scientific method" depends on how that phrase is defined. To the best of my knowledge, it does not have a standard definition, so that question cannot be answered.)

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    "[in mathematics] there is no such thing as proof". Um, I'm not downvoting, but proof is a major cornerstone of mathematical thinking. See en.wikipedia.org/wiki/Proof_theory
    – J D
    Mar 14 at 20:44
  • There is a lot of people, including mathematicians, saying Math isnt a science but a language
    – Pablo
    Mar 14 at 21:50
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    Do you know the story of the six blind men and the elephant? Each man thought the elephant was like its tusks, or its legs, or its ears, or its sides, or its trunk, or its tail — not realizing the elephant was all of those things. (Oh, and can you please name one actual mathematician who is recorded as having claimed mathematics is not a science, and where this is stated?) Mar 15 at 0:05
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    J D: The phrase "there is no such thing as proof" applies to the noun phrase closest to it: "other sciences". Mar 15 at 0:09
3

Mathematics is a tool that can be used in several ways:

  1. Axiomatically as in finding new theorems and conducting reverse mathematics to find new axioms.
  2. Scientifically for modeling physical phenomena as in using math to support inductive claims, such as statistical sampling or to describe physical phenomena, such as in quantum statistical mechanics.
  3. Computationally when large systems are used to determine answers to difficult algorithms or when proofs are automated.
  4. Approximately as when ordinary diffeq is too difficult to compute, and approximate answers are calculated using numerical analysis.
  5. Empirically, such as in computer science, when mathematical formalisms are implemented as programs, executed producing complex results, and then are analyzed to see what happens. Consider how the primes as generated by the Sieve of Eratosthenes aren't algebraically determinable.

So, it's fair to say these methods are scientific, but the are not the 'scientific method of explaining the natural world' which is a bit of an abstraction of the various scientific methods of various sciences. But if we accept the general abstraction, we can note that both mathematics and science are forms of problem solving:

  1. Identify the problem (Observe phenomenon/Establish claim to be proved)
  2. Propose a solution (Propose hypothesis/Use techniques to infer towards claim)
  3. Test a solution (Test hypothesis/Prove lesser claims)
  4. Revise approach until solution (Continue until strong theory/Systematically show claim is established and consistent)

But they differ in several important regards.

  1. The scientific method is strongly aligned with the correspondence theory of truth. Science wants to know that language matches physical reality. Mathematics tends to be much more interested in logical consistency, and is therefore closer to the coherence theory of truth.

  2. Mathematics is largely a rational activity. It has to do with thinking through claims and ensuring that language functions properly. It's fair to say that mathematicians' bread and butter is the application of reason to the mathematical ontology of abstract objects (SEP). Science on the other hand, philosophically speaking, is largely empirical. You'll hear words like observation, testing, and sensation thrown around.

  3. Mathematics tends to attract and be used in a deductive fashion. It tends to be very focused on deductive strategies within the abductive context of problem solving. Science on the other hand solves problems abductively by emphasizing the inductive aspects of reason. There's much more a willingness to confront theory with an understanding of defeasibility (SEP).

So, no, math and science, while in the broader spirit of scientific thinking are both systematic application of problem solving, are very different beasts, and tend to attract, by my personal experience, very different personalities. Albert Einstein himself famously considered himself a poor mathematician. Mathematical methods, may be the queen of the sciences, but are usually considered quite distinct from contemporary scientific methods.

2
  • Given Johan's comments elsewhere, I think it's fair to point out that math can be done inductively (as in computer science where we write software which is extraordinarily complex and then see what comes of it at run-time). Reverse mathematics is also very non-deductive. But what makes math truly distinct from science is its non-empirical domain of discourse.
    – J D
    Mar 13 at 16:12
  • May be of interest: en.wikipedia.org/wiki/Quasi-empiricism_in_mathematics
    – J D
    Mar 14 at 13:40
3
  1. In common language science is considered the science of nature, i.e. a certain discipline which deals with phenomena in the animated and non animated nature. In that sense mathematics is not science, mathematics does not rely on observation and experiment.

  2. Mathematics deals with ideas, and these ideas are human-made. Mathematical concepts are free creations of humans. The characteristic of mathematics is its deductive method: Fixing axioms and fundamental definitions and making deductions, i.e. proving theorems by following rules of logic.

    Mathematical proofs provide true statements, but that has its price: The domain of mathematics is totally independent from the ambient real world.

  3. It seems to be a miracle—at least there is no general accepted “why?”—that mathematics nevertheless serves as the most useful language for science.

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    +1 For concision. I"m not sure that it's quite miraculous. Survival fitness requires an organism to both understand and explain things (science) and count and track the motion of thing in space (mathematics). It's not good enough to know lions are dangerous and might eat you. One also has to count them and know where they are around you and if they are moving towards you.
    – J D
    Mar 13 at 15:44
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    Then, why does the last link I posted talks about "Using the Scientific Method to Engage Mathematical Modeling: An Investigation of pi" ? Aren't they using the scientific method and claiming they do, or what?
    – Pablo
    Mar 13 at 15:53
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    Also, for example, when you try to a demostration in Mathematics, don't you take an equivalence or something, replace things here and there, and see if you reach the desired result? Isnt that substancially the same mechanic the scientifc method use, in the scientific method you do an experiment where you isolate and introduce a change, and see if that gives you the result? Regardless what characters represent, in the end you are going to visually "empirically" observb if the equivalence and replacements you used end producing the result /characters / formula / equation you were looking for
    – Pablo
    Mar 13 at 15:58
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    When you can manipulate Math in a totally independent way of the real world (I remember a mathematician saying his work was to solver either equations or integrals sent by engineers, and that he had no idea what they were for), the fact that their rules emerged from the real world, does that make it totally independent? I mean, 2+2 = 4 and not 5, and there has to be 4 things of something somewhere if you add 2 things, their rules were meant to represent the real relation between adding 2 things, and it wasnt an arbitrary made up rule in that sense
    – Pablo
    Mar 15 at 2:39
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    @Corbin I think modern number theory is number theory after Grothendieck’s introduction of schemes, today termed Arithmetic Geometry. But of course Dijkstra is right: Powerful computer programs like Pari are used as a tool when working with subtle ideas in number theory.
    – Jo Wehler
    Mar 15 at 16:41
3

Let's start with an informal description of how science 'works' which isn't exactly a formal explanation of the 'scientific method'.

  1. Start with some assumptions, a hypothesis
  2. Based on those assumptions use logic to derive measurable predictions
  3. Design an experiment based on those predictions
  4. Execute the experiment and evaluate how well the results match the predictions

With this process complete, conclusions can be drawn. Assuming the logic used to derive the predictions is sound, if the results do not match reasonably well with the predictions, the scientific answer is that there is a problem with the assumptions. They could be completely wrong, partially wrong, or they might be incomplete. This is the goal of science: to show what assumptions are incorrect.

Mathematics starts with axioms. These are like our scientific assumptions but with a pretty significant difference: they cannot be disproven. What I mean by that is if you start with the axiom that 0 is a natural number you can't disprove that 0 is a natural number. (You could find some sort of logical contradiction or inconsistency but that's a rabbit hole I am not going down.) By definition, axioms are definitional.

My take is that this is the fundamental difference between math and science. Science seeks (ultimately) to reveal the 'axioms of nature' through a process of elimination. Mathematics is the exploration of the implications of axioms.

However, that doesn't mean that you can't use a similar approach within mathematics. We can, for example, start with the assumption that all prime numbers are odd, then start listing prime numbers and evaluating (based on our axioms) whether they are odd and quickly find that the assumption is wrong. But the idea of building a 'mathematical experiment' to try to prove your axioms wrong makes no sense. It's axioms that ultimately determine whether a mathematical statement is wrong.

Back to the question: does mathematics use the scientific method. Not really, but sometimes approaches that are similar (if not essentially identical) to scientific processes are used. I would also add that it's pretty important to science that mathematics not be a science. Our approach to science would be on very shaky ground if we can't be sure whether 2 + 2 equals 4.

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    I would precede your four points with 0. Make observations and measurements of your subject of study. Mar 15 at 1:04
  • Your closing comments need to be inscribed deep
    – Rushi
    Mar 15 at 1:46
  • @Rushi That video suggest you might be missing my point. Physicsts are free to choose different kinds of math (e.g., hyperbolic geometry versus Euclidian.) Sometimes they invent their own math. But if the math was being 'tested' along with the hypothesis, it doesn't really work.
    – JimmyJames
    Mar 15 at 20:37
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    @JimmyJames Precisely! The physics is being discovered by whatever process. The math too may be discovered by some process. But if the 'experimental lab' so to speak where these emerge are the same we have a problem
    – Rushi
    Mar 16 at 1:16
  • This answer made me think about how the question depends so heavily on what we mean by “the scientific method”. I personally think it’s a cliche concept. We need a much deeper analysis of if there is any unifying characteristic of “science”, something like “empirical knowledge”. Apr 6 at 15:02
3

The existing answers don't seem to take into account the role of Platonism in this issue. From the viewpoint of Platonism, and/or if one emphasizes the idea that mathematics is a "formal science" (whatever that means), mathematicians study a mind-independent reality which therefore has nothing to do with the empirical method of the natural sciences. On the other hand, other schools of thought will obviously find more similarities among the exact sciences. So to ask a more meaningful question, one would have to specify which school one is interested in.

In this context, it may be interesting to examine the position of Marburg neo-Kantianism led by Hermann Cohen around 1900, including Ernst Cassirer who was one of his most brilliant students. From the point of view of the Marburg school, similarity between the methods of the natural sciences and mathematics was always emphasized. A related study can be found here:

Mormann, T.; Katz, M. "Infinitesimals as an issue of neo-Kantian philosophy of science." HOPOS: The Journal of the International Society for the History of Philosophy of Science 3 (2013), no. 2, 236-280. http://doi.org/10.1086/671348 and https://arxiv.org/abs/1304.1027

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  • It's hard to think Math as totally independent of the real world when numbers appeared to represent quantities. If Math would have appeared as something totally disconnected from the real world, there would be no reason for 2+2 to be 5, 6, or whatever
    – Pablo
    Mar 17 at 20:51
  • @Pablo, I take it neither of us is a Platonist :-) Mar 18 at 9:53
2

No, for two solid reasons:

First, the core formal system of all formal systems is Logic (sustained tautologically by itself). Upon it, there's Mathematics (if it's not there, it's near there). Upon it, the rest of systems: the scientific method is way farther on the stack.

...
n. Scientific method (it depends on empirical facts, it is strictly not a formal system)
2. ...
1. Logic (this is the core formal system, circular, tautological)
0. Reason (it is not a formal system, it is not necessary logical)

So, it is a fallacy to think that Math depends on the scientific method, because it is the opposite: any overloaded* system/method depends on Logic and Mathematics.

Second, in addition: Mathematics is essentially metaphysics (for example, the number 12, an equation, a sphere or a matrix are ideals, not physical facts). So, while Mathematics targets metaphysical truth, the scientific method targets science, which targets empirical truth.

If you don't know the difference, philosophically, there are two fundamental domains of knowledge: Aristotelian physika and meta ta physika. What is physical is what is related with the senses (yes, the five senses), that is, what we can know of the physical world. Conversely, what is metaphysical is what is pure, rational, independent of the senses.

So, while Mathematics is metaphysical knowledge, the scientific method is related with empirical knowledge (remember this: science targets empirical truth). So, by definition, and by fact, Mathematics can't use the scientific method.

* See Semantic Overload.

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  • I would not classify mathematics as metaphysics: Mathematics is clear and precise, while metaphysics is obscure and cloudy :-)
    – Jo Wehler
    Mar 13 at 15:04
  • @JoWehler 1) Your conception of metaphysics is the popular synthetic a posteriori metaphysics (e.g. God, Aesthetics, Ethics...), which is a mix of physics and pure metaphysics. I refer to PURE metaphysics as in the Critique of PURE Reason. 2) Metaphysics is not necessarily obscure, and if so, the same happens with physics (e.g. quantum gravity, strings theory, etc). 3) If Mathematics is not meta ta physika, it must necessarily be physika, which is not. Mathematical entities are purely rational ideals (metaphysical), not observable facts.
    – RodolfoAP
    Mar 13 at 15:15
  • To whom it may concern: I do not understand why Rodolfo's answer has been downvoted in short time. One can question to consider mathematics as metaphysics. But better than just downvoting were presenting some arguments like the responder presents for his/her viewpoint.
    – Jo Wehler
    Mar 13 at 15:29
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    I didn't downvote, but I too reject the claim that metaphysics and mathematics are essentially the same. Metaphysics has the primary preoccupation of providing first principles of reasoning. Mathematics is the study of quantity, relations, operations, shapes, and directions. I'll upvote because there are sensible claims here overall.
    – J D
    Mar 13 at 15:40
  • @JD In effect, Metaphysics and Mathematics are not "essentially" the same. But Mathematical knowledge is predominantly metaphysical.
    – RodolfoAP
    Mar 13 at 15:54
2

The paper you linked is not about math but about "science education" and about "mathematical modelling". This of course is very much at the core of the scientific method: Build a model of the real world (usually expressed in the language of mathematics) and then compare the predictions of this model with what real measurements show.

As for math itself: it is a language. A set of definitions and rules of how to manipulate symbols. It can not be "right" or "wrong". As a language it is only useful if it is (at least potentially) understood by others, else you can not use it to communicate.

For sure you can "invent" your "own kind of math". A definitions can be useful - which means it has the potential to be applied well to communicate certain ideas or model certain aspects of the world, or it can be arbitrary and useless. A system of definition can be contradictory. E.g. by having the rules not carefully defined you could quickly "proof" that 1+1=3 and from this you could then quickly "proof" anything. Such a system of definitions which allow you to "proof" anything then of course allows you to proof nothing. So it would fall under the "useless" category of definitions.

And then: There is actually some "experimental" character in the math of today. E.g. to disproof a statement you only need to find one counter example. And often people use computers to try to find them. So this has a vibe of "experimental" to it. If I do some symbolic calculation and want to check them I usually run some numeric simulations to test if the symbolic results are correct or if I have an error there. If the numeric result agrees it is not a proof but if it would disagree then I know I need to check things again.. so also some "experimental" character..

2

Plato himself, in the dialogue “Meno,” purports to offer experimental evidence that souls recollect mathematical facts from their pre-existence in a world of abstract Forms.

In modern times, some mathematicians, notably Gödel and Quine, have argued that mathematical research should be extrinsically motivated, for its ability to accurately model the universe. This is the basis for Putnam’s “Indespensibility argument.” The Stanford Encyclopedia of Philosophy defines methodological naturalism as the belief that “the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural,” in its article on naturalism in the philosophy of mathematics.

A widely-cited contrary view is Paul Benacerraf’s 1973 paper, “On Mathematical Truth,” which argues that it is impossible to apply our principles of epistemology to purely abstract objects.

2

What can be described as science lies on a spectrum.

You can have subjects where empirical observations are 'easy' to produce and theories are tested against them regularly and change accordingly, e.g. physics.

You can have subjects where empirical observations are 'hard' to replicate and we have a much harder time agreeing on when to update our theories, e.g. economics.

You can have subjects where empirical observations are 'impossible' to replicate, the realm of the idiosyncratic where every observation is unique and it's very hard to agree on theories at all, e.g. psychoanalysis.

Then, you have a subject where empirical observations are 'trivial' to replicate in the sense that they follow deductively from a set of rule and axioms and they always turn out to produce the same identical outcome and there is no need to update the theory, mathematics.

And yes, I argue that these are all science, to a degree.

2

Mathematical modeling will always follow or use important aspects of the scientific method. A mathematical model is created to describe behavior of a system (physics, chemistry, etc) using mathematical equations. Since the purely mathematical system is modeling "scientific" behavior, the scientific method must be used to verify the accuracy of the model against "real world" results.

Geometry is an example of mathematical modeling. The equations of ideal shapes that model shapes seen in nature: the moon, quartz crystals, etc.

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    The formal proof is only the "reporting conclusions" part of the scientific method. You just don't usually know the methods and failed proof attempts that came before it.
    – Mathaddict
    Mar 14 at 22:10
  • 1
    +1 @Mathaddict Niels Abel once said of Gauss, "He is like the fox, who effaces his tracks in the sand with his tail." to which Gauss replied, "No self-respecting architect leaves the scaffolding in place after completing his building." More bluntly math proofs are more lies than proofs as to the actual discovery process
    – Rushi
    Mar 15 at 5:55
  • @Rushi You are correct. I've removed all references to formal proofs. Thanks. Mar 17 at 16:51
  • @Mathaddict You are correct. I've altered my answer to remove references to a formal proof and focused on mathematical modeling. Thanks. Mar 17 at 16:52
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Both math & science are, at heart, logical projects. There's deductive logic (math) and there's inductive logic & abductive logic (science).

Deductive math has a date of origin - when Euclid began writing his The Elements. However, the knowledge of The Pythagorean Theorem predates Euclid by not a few, but thousands of suns (could the pyramids be built without knowledge of a² + b² = c²?).

The geist of ancient math survives ... the Goldbach conjecture has been tested on a bazillion even numbers. The black swan hasn't been spotted yet ... yet!!

1
  • i thoughjt this was a good answer, but it seems that ancient egypt was aware of pythagorean triplets, not the theorem in full. was that artistic license?
    – andrós
    May 17 at 4:32
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Math does follow the scientific process. For example, the oldest mathematical discipline is arithmetic. This was originally a physical theory about individuals. It was at this stage that it was empirically validated that 1 +1 = 2, that it was commutative and associative - even if not explicitly stated as such. It was the first physical theory to be made formal. Math and physics are not two separate disciplines but show a dialectic between them.

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  • I find it absurd that someone would claim 1+1=2 (or any other mathematical statements) can be validated empirically. What kind of experiment can be done to validate such statements? Did people use such experiments to determine the validity of statements? What does it even mean to validify a mathematical statement empirically?
    – Poscat
    Apr 6 at 14:52
  • @Poscat: One drop of water plus another drop of water is not two drops of water ... Apr 6 at 17:53
  • @Poscat: the essentials of arithmetic, as an empirical study, is so obvious that most people do not consider it empirical at all. Apr 6 at 17:54
  • 1
    How does that has anything to do with my questions?
    – Poscat
    Apr 6 at 23:38
  • @Poscat: One drop of water plus another drop of water is not two drops of water. But one tree plus another tree is two trees. Hence addition is empirical. Hence a science as umderstood as an empirical field of study. Apr 7 at 18:57
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Does math use the scientific method?

No. Every scientific theory must be falsifiable, i.e. we must be able to specify experimental or observational data that would disprove it. The theory of evolution, for example, would be falsified or disproven if we could be prove that every species that have ever lived on Earth once co-existed here.

There is no such requirement for mathematical theories. They require only internal consistency, or at the very least, an absence of any known inconsistencies after intensive study.

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  • 4
    Not downvoting, but not all math is constructing math theories. Mathematical model building is definitely an empirical exercise. Just ask the IPCC.
    – J D
    Mar 13 at 22:52
  • 1
    @J D Disagree that math model building is true math, i.e. pure math.
    – Trunk
    Mar 13 at 23:01
  • 2
    Obviously, mathematical statements are falsifiable? Trivially: by finding a counter-example, and less trivially, by exposing a flaw in the reasoning. Mar 14 at 9:58
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    @Peter-ReinstateMonica Can you give an example of a counter-example to a mathematical theory? Perhaps you mean an "inconsistency" in the style of Russell's Paradox? Note that "false" is not the same as "inconsistent." Mar 14 at 15:40
  • 2
    @DanChristensen mathoverflow.net/questions/35468/… has a plethora of them. Most are above my pay grade, but apparently the Jacobian Conjecture was considered proven until somebody came up with a counter-example. I'd also like to note that "statement" is a wider term than "proof"; many "statements" (probably called "conjectures" by the peers) were disproven, usually by counter-examples. Mar 14 at 16:03

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