Agreed, and as such, mathematics rightly belongs to the colleges of natural science. Are you trying to conflate natural science with liberal arts? I might be confused by your comment.
He is trying to erase the distinction when talking about mathematics specifically because it leads you to question your own biases.
Mathematics is neither liberal arts nor natural science.
Mathematics is a liberal art.
Mathematics is a natural science.
Mathematics is both liberal arts and natural science.
Which is the “correct” classification of Mathematics is the sort of pointless quarrel that happens at universities.
There is no right answer - the purpose is to argue and understand all biases/positions. The purpose is respectful human interaction around unresolvable differences.
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.
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.
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.