Everytime a new area of maths is "invented" an application in the physical world seems to soon follow that suggests that the natural world is conforming to some mathematical laws whether we are aware of that mathematics or not.
Even the simple arithmatic example you give is far from definitive. Do we really think of the product of 3 and 4 as purely abstract or as 3 somethings multiplied by 4 of the same something to end up with 12 in a very physical sense?
I don't know the answer to whether or not maths is invented or discovered but I don't think it is as clear cut as you suggest
I am more interested to know if people think we doscover or invent maths?
Kronecker famously said "God gave us the natural numbers. All else is the work of man." Personally, I think Kronecker gave God too much credit.Compassionate conservatism - bringing you a kinder, gentler torture chamber
At first blush, I would tend to have already said that I think I would "discover" science but "invent" math.
I'm not so sure you can easily separate discovery from invention. There is a lot of invention in science, and there is a lot of discovery in mathematics. Sometimes we invent in order to discover.
Maybe we invented the natural numbers. But mathematician discovered the prime number theorem. Sure, if they invented numbers, then the prime number theorem is a consequence of that invention. But it is not an obvious consequence, so required discovery.
In science we invent methods of getting data, in order to discover things about the world.
It is nice to notice someone suggesting that I present something "clearly" here. You asked and I answered.
Sure it seems to be a case that once a "new area" of math appears in culture the world can sublimly become descripted to "conform to some mathematical laws" even though a collective "we" of the world are not aware of (it) or not.
I would just see that as one stage in the ever expanding "tool set" of mathematicians as to when it might become true that some other math is developed that can do a better job of informing the same formation of lawlike behavior at a second remove.
I am actively thinking of how transcedental numbers can gain say the patterns findable in evoluiontary theory but as of yet there are no "laws" even though the math already exists. If I was to write these applications the older math of "parents" as expressed in population genetics would walk the rope of finer line so constructed. In that future, I might be aware of subjective elements that pass for nothing but a stage in my personal horizon only later to be upbraided (if true in your sense).
When I THOUGHT the product I thought of it NOT as I FIRST LEARNED IT, as a rote table but as something with a potential sense in population thinking and thus "abstract" but via an application rather than a formal 'table' instantiated in a form that might also be analogous no matter the application but as the application was about homology the math and the bio-physical sense were seperated as to the normal form the logic of it would detail.
Using my own ideas is not going to be useful as this does depend on the factual truth which only the math and not my thought of it depends. One could of course say something different if one was refering specifically and only to past episodes in the mathematical history. That is what I meant by giving it a harder and second thought not this explanation of my first thought or sequence from a given thought.
Math is the one part of doing science that is as clear cut as I suggest.
Personally I'd say an Art with the personality of a Science. A very different intellectual urge is satiated when you do maths to when you do science.
The most mysterious thing about mathematics is how useful certain, seemingly abstract, areas of it are. For instance the fact that certain techniques in Algebraic Topology, invented solely for classifying spaces in pure mathematics have a use in modern particle physics is very bizarre.
Although I'm always impressed by how the different areas are used in proofs from other areas. For instance there is an analytic, topological and algebraic proof of the fundamental theorem of algebra. That connectedness between branches of mathematics that were conceived for entirely different reasons is what I find most unusual, probably more so than its applicability to science.
Perhaps I just have not "discovered" enough math. I dont dispute that there can be a humane process that moves from appearences of discovery to invention to discovery. Yes, I dont doubt the history of math to be able to uncover some such.
With math rather than a particular disciplie of science, I can not restrain myself from feeling that SHORT of "discovery" I am simply 'ignorant' and thus, if I was to get "beyond" that state of mind, I would need to "invent" a way beyond rather than simply feel I was beyond no matter what discovery would bring. I do not feel this way in some areas of science. I can feel very certain that NO MATTER THE DISCOVERY it would not matter what I could invent. I can gain a fairly clear sense that no matter what I do not know it is not because I am ignorant. This I can not do when I come to the highest level of "mathematical maturity" I can pretend or think I have. I always feel in math there is a collosal future before that. In science I just have a horizon infinite in two directions, to say it shortly.
Could we prove that math is not science by stating the obvious fact that math is but a tool used in science, just like a bleeker?
There is a branch of philosophy, known as "epistemology", which supposedly studies the theoretical principles of knowledge. In my opinion, much of it is silly. I favor the view that mathematics is the real epistemology.Compassionate conservatism - bringing you a kinder, gentler torture chamber
As a scientist, was wondering this myself! The only thing I could think of was Beaker, as in Dr Bunsen Honeydew's rather tortured assistant :) Perhaps Beaker got his name from Bleeker, or was it the other way round???