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>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehement

>Today, mathematics is regarded as an abstract science. Pure mathematics is regarded as an abstract science, which it is by definition. Arnol'd argued vehemently and much more convincingly for the viewpoint that all mathematics is (and must be) linked to the natural sciences. >On forums such as Stack Exchange, trained mathematicians may sneer at newcomers who ask for intuitive explanations of mathematical constructs. Mathematicians use intuition routinely at all levels of investigation. This is captured for example by Tao's famous stages of rigour (https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...). Mathematicians require that their intuition is useful for mathematics: if intuition disagrees with rigour, the intuition must be discarded or modified so that it becomes a sharper, more useful razor. If intuition leads one to believe and pursue false mathematical statements, then it isn't (mathematical) intuition after all. Most beginners in mathematics do not have the knowledge to discern the difference (because mathematics is very subtle) and many experts lack the patience required to help navigate beginners through building (and appreciating the importance of) that intuition. 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. The mainstream view in mathematics is that infinite sets, especially ones as pedestrian as the naturals or the reals, are not particularly weird after all. Once one develops the aforementioned mathematical intuition (that is, once one discards the naive, human-centric notion that our intuition about finite things should be the "correct" lens through which to understand infinite things, and instead allows our rigorous understanding of infinite sets to inform our intuition for what to expect) the confusion fades away like a mirage. That process occurs for all abstract parts of mathematics as one comes to appreciate them (expect, possibly, for things like spectral sequences).

pdpi 1y ago
> 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.
- 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
ndriscoll 1y ago
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-does-an-open-set-axiomatise 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.
nkrisc 1y ago
> 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.
- empath75 1y ago
  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.
  - kannanvijayan 1y ago
    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.
    - mrguyorama 1y ago
      Hell, 0 used to be considered too abstract!
    - griffzhowl 1y ago
      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
gaze 1y ago
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.
griffzhowl 1y ago
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 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 https://www.youtube.com/watch?v=M-MgQC6z3VU

pdpi 1y ago

> 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.

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

ndriscoll 1y ago

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-does-an-open-set-axiomatise 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.

nkrisc 1y ago

> 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.

empath75 1y ago
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.
- kannanvijayan 1y ago
  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.
  - mrguyorama 1y ago
    Hell, 0 used to be considered too abstract!
  - griffzhowl 1y ago
    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

gaze 1y ago

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.

griffzhowl 1y ago

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 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 https://www.youtube.com/watch?v=M-MgQC6z3VU