Totally disagree. Mathematics is not a science at all because there is no any possible experiments in the Math fields.
And remember what David Hilbert has told about one of his students who decided to go into arts instead of keep learning Math with him. He told "he did not have enough imagination to become a mathematician".
Experimental mathematics is not a field of mathematics, but an approach. In the link you have given I found two approaches called experimental mathematics: proofing theorems by computer contrary to proofing with a pensil, and searching new problems contrary to solving some existing ones.
I was responding to your apparent claim that experimentation is not possible in mathematics. Maybe I misunderstood what you meant.
From Halmos' quote from the wiki page:
"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. You want to find out what the facts are, and what you do is in that respect similar to what a laboratory technician does"
- it sure sounds like he thinks experiment/experimentation a key part of maths.
What evidence would convince you that mathematics is empirical?
If your argument was a tautology then what’s the point?
Would you consider interactive theorem proving to be empirical or not? The sort of experiment you perform is you try to see if the computer accepts your proof or not.
It sure seems like the goal posts are being moved.
Every experiment is an experiment of nature.
Humans are part of nature.
Human constructs are part of nature.
Every interaction that results in the testing or falsification of some hypothesis is a valid experiment. Irrespective of the object being experimented/interacted with.
I am experimenting with (testing/falsifying hypotheses against) your linguistic constructs right now.
I think you're missing the point. Mathematics isn't a natural science, and that's what I was trying to point out. There is a term for doing science on human constructs - it's called social science. Is your thesis that mathematics is a social science?
> Humans are part of nature. Human constructs are part of nature.
Humans physically are part of nature, sure. Human constructs are not considered to be as far as classification in the social and natural sciences.
I am not sure why you are using the phrase "natural science", is there any other kind? Lets revisit with the wikipedia definition:
Science (from Latin scientia 'knowledge') is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
That seems pretty general and all-inclusive to me. Any knowledge about humans; or societies; or the stuff human societies invent is knowledge about the universe. Because we are part of the universe.
Now, you seem to be trying to differentiate the different sciences from one another - tell us why. What's the purpose of drawing a distinction between science and (what you call) "natural science"?
I am not doing so. I am saying that if you want to classify mathematics as a science on the basis of the existence of theorem validation software on which someone may experiment, then it is a science on human constructs, ie, a social science, and not a natural science.
The argument here for math to be a science is very tortured and contrived, but if it has any even technical validity then it makes the argument for math as a social science.
>if you want to classify mathematics as a science on the basis of ... The argument here for math to be a science is very tortured and contrived,
I am not classifying Mathematics as a science on the basis that you claim. I am not even making an argument! All I am doing is empiricism!
I am observing a fact about the universe: What humans call science is any enterprise which builds and systematically organizes knowledge about the universe. I am also observing that Mathematics is one such enterprise. So it's a science.
If you think society's definition of "science" is too broad - that's fine.
If you want to use a narrower definition which renders Mathematics a non-science - that's also fine.
But I am still asking why?
Why does it matter whether Mathematics is classified as a science; or a non-science; or a social science; or a natural science?
I disagree. People haven't had a Mathematics before Pythagoras. All Mathematics people had at that moment was that 1 golden coin + 1 golden coin = 2 golden coins or one copper knife. But trading is not mathematics if not talking about modern things like HFT with broad using of probability theory (and we obviously are talking about ancient times).
Physics is understandable even with way more simple (and way more ancient) species then humans. For example, an ape can take a stick and use more heavy end to make a more powerful hit. Birds seems to understand physics quite well - starting from setting wings while flying and ending to general intelligence of crows.
And BTW why are you concerned that in the pyramid of knowledge (Biology is a science > Chemistry is a science > Physics is a science > Mathematics is a language) there are three sciences and one a non-science?
Mathematics is a language, but then why are you saying there was no Mathematics before Pythagoras? Obviously Pythagoras didn’t invent language! Long before Pythagoras people had been using language for doing and expressing computation.
Pāṇini did it. The Babylonians did it. It then took us a few thousand years to mechanize that knowledge and invent computers as we know them.
As for physics being “understandable” - I think you are conflating the ability to exploit nature with the ability to understand it.
Physics isn’t something “out there”. Available to birds and apes. It it is the body of knowledge (texts, ideas, concepts, formulas, narratives) instrumental to humans navigating nature.
There would be no physics without humans inventing it just like there would be no language without humans inventing it.
I am not at all concerned about the contents; or the structure of the pyramid. I am simply expressing a fact about the pyramid. That which people call science is not founded upon science. It is founded upon the liberal arts, humanities and social constructs (logic, philosophy, mathematics, computation, language) - the pyramid is founded upon human invention.
> Mathematics is a language, but then why are you saying there was no Mathematics before Pythagoras? Obviously Pythagoras didn’t invent language!
He did invent the language. When Pythagoras used to teach some folks the division (mathematical operation required for Trigonometry which is required for building Egyptian Pyramids) he faced with un-understanding of "why we should learn this silly useless numbers?" Then Pythagoras has invented a music (consonance and dissonance at least) and everyone started to learn Math because most of us can hear that numbers which may lead to either consonance or dissonance.
> you are conflating the ability to exploit nature with the ability to understand it.
Yes you are right, I do not see how can you understand Physics/Nature without exploiting anything. And if you are enough successful to exploit something so why not to claim that I have gained some understanding.
> There would be no physics without humans inventing it just like there would be no language without humans inventing it.
I think two instances of "humans" word are extra, that words don't add anything into the discussion of some mathematical and physical concepts.
And I keep staying on my opinion that: if one fly can do something complicated in controllable experiments set by human (kind of pull this rope and get some sugar) and another fly can learn this queue of actions from the trained fly, that means that after the experiment both flies share some physical knowledge about Nature.
>Yes you are right, I do not see how can you understand Physics/Nature without exploiting anything
I am happy to equate exploitation with understanding, but then I don’t see how you could possibly understand Mathematics without exploiting computation.
The pyramids were built at least 1000 years before Pythagoras was even born. If division and trigonometry was needed to build them then the Egyptians understood and exploited division trigonometry before Pythagoras invented the language to explain/teach division and trigonometry.
It is not at all surprising that nobody else could understand (exploit, compute with) a language Pythagoras made up, but eventually he taught (indoctrinated?) some students on how to use his invention.
To compute with; and exploit any language first you must comprehend its grammar and semantics.
Still! The Babylonians could compute the square root of 2 (in practical terms, obviously they didn’t call it “square root of 2”) to arbitrary precision even without having numbers in their Mathematics.
Overall I don’t think we are disagreeing over anything substantial either way. Computation is the controlled manipulation of matter. Manipulating symbols to do useful things is a form of computation.
A fly may have some knowledge sufficient to exploit nature in the moment, but it doesn’t have the knowledge necessary to encode and communicate its knowledge to its peers. And it certainly can’t communicate its knowledge to generations 2500 (or more) years into the future.
The point about Pythagoras is that he made Math really popular and maybe even enough broad to use it for Liberal Arts. The Egyptians which have build pyramids used to keep all their Mathematics in secret, only priests and some high-rankings could learn from their sources - and for limited set of goals.
What about Babylonians, their math (as well as Egyptian's) were too complicated for writing and (unless Greek's) was experienced some lack of integrity. For example, did Babylonians know that the square root of 2 is irrational? Pythagorians knew that exactly.
For me it is much more handy to consider well-known Pythagoras as father of our omnipotent Mathematics instead of some anonymous Babylonian who does not have an epic story about stealing some knowledge from totalitarian company and presenting it to mortals like Robin Good or Prometheas. And in that formulation I consider omnipotent Mathematics to be younger than Physics which even now has not achieved omnipotency level (hint about Theory of Everything).
And sorry but I have nothing to say should a CS be a part of Liberal Arts.
Any experiment has an abstract result. For example: If throwing a rock from the tower of Pisa gave us result that acceleration of gravity on Earth is 9.8 m/s^2 then that number will be exactly same for throwing any other materials from any other tower on the Earth.
I feel like we have a better definition of math than we do of art. Thus it's a lot easier to accurately declare a thing is not math, than it is to accurately declare a thing is not art.
Every other field of study has more artsiness than Mathematics.
However, Mathematics also still has a lot of artsiness to it.