Mathematics describes
truths. In this, it is almost
unique. If I have a
theorem, what it says is
unconditionally
true! Of course, we might be in
error, and think that some
proposition is proven, whereas in fact it isn't. But that's unavoidable in any
human endeavour.
Even physics cannot lay claim to this position. Physics is (for the most part) a description of observed reality. As such, a physical law may be wrong, even though we have made no errors. Our measurements may not have been accurate enough, or we may not have measured the right thing, or any of the other problems which plague an empirical science may have cropped up.
Of course, mathematics is performed by humans, who are (for the most part) social beings. And the mathematical tools we develop influence the observations we can make of the real world. But our mathematical tools always have a privileged position: they are TRUE.
Tem42: "True within a system?" What "
system" exactly is being used to describe mathematics?
And physics describing the world? Physics gives you (systematic) empirical knowledge of the world.
Physical knowledge is subject to empirical validation; when that fails, we have to change our physical knowledge. For instance, when Newtonian mechanics turn out to be insufficient to describe the world, we need to switch to relativistic physics and quantum mechanics.
But look at this amazing coincidence: All 3 theories are firmly based in mathematics! And NOTHING in the mathematics used to describe the newer theories which supercede Newtonian mechanics invalidates the mathematics used in the older theory. How come?
Quite simply, the mathematics is correct. Unconditionally. The sum of the angles of a triangle in the plane is 2 right angles. This is the sort of mathematical knowledge you need to describe classical physics. Note, however, that on a differential manifold the sum of angles of a triangle is not 2 right angles. For instance, on a sphere it is always strictly greater than that. Elaborations of this sort of knowledge are used in general relativity.
But here's the really cool bit: NOTHING in the theory of differential manifolds says that the sum of the angles of a triangle in the plane is not 2 right angles. How could it? It's TRUE. So one physical theory has replaced another (which is now considered "inaccurate"), but the mathematics is still perfectly good.
This also explains the mathematical character of physical law. Physics does all the dirty work of explaining empirical reality (that is, "what happens to be", as opposed to "what must be"). Naturally, what's left afterwards (to formulate the precise law) requires an absolutely true language -- mathematics.
Finally, a challenge to anyone who holds math to be a social construct: present an example. That is, show a bit of mathematical knowledge that is valid in one social setting and not valid in another social setting. That's all you have to do to convince me...
Muke: Your first example is very clear. Indeed, one
cannot define
division by zero consistently and retain the
field structure of the
real numbers. One
can define division by zero by losing some of these properties (e.g.
IEEE floating point numbers), but one is no longer working in the real numbers. What is the
social construct here? The
Heracles example, however, is simply incorrect. I do not understand how Heracles
divides his
6 cows to
nobody. There are certain properties that we would expect to hold when we say Heracles divides his cows. For instance, that each cow is given to someone (i.e. the division is in particular a
partition of Heracles' cows). And if we accept these properties, then Heracles cannot divide his cows to nobody (since the
empty set is not a partition of any non-empty set). This is one way of defining what we mean mathematically when we use the term "divide". You seem to hint at another possible mathematical interpretation of the word "divide". So you think there is more than one mathematical interpretation of "divide".
Obviously you must agree on an interpretation. But this is merely because natural language is
imprecise (among other things, it allows you to say things like "divide the cows to nobody" which are scarcely clear). If you look at something like the
Monty Hall problem or the
two envelope paradox, you get even better examples of imprecise or misleading phrasing in natural language. In all
3 cases, after removing ambiguity, nothing social remains. So if this were all, we could replace "math is a social construct" with "natural language is ambiguous", with which I would gladly agree.
Regarding Torddjen, that is a non-example. Unless I've done the math wrong, after going through the motions you describe, Torddjen now has 0 lasers (or "has no lasers", if you prefer to avoid zero too), and he owes !Kou 6 lasers and Ko'lei 2 lasers. This information is stored in MegaHAL's databanks, too.
The mathematician in Torddjen might want to know if he should get more lasers. After asking MegaHAL "how many lasers have I got, and how many do I owe", Torddjen sees that he has no lasers and owes 8. Since 0<8, Torddjen makes a note to head to the pawnshop and buy 8-0=8 lasers (working without 0, he'd simply skip the issue of the lasers he owns once he discovered he doesn't own any). Strapped for cash, Torddjen recalls an old gambling debt of 11 Adranian pints of Borgolias whiskey, drinks 5 (the recommended daily dose) and is left with 6 to pay at the pawn shop.
Yes, it would be more convenient to work with negative numbers. No, Torddjen would not have to change his business strategy, as he would owe the same number of lasers and have the same number of pints of Borgolian whiskey available to him. This is amazing! Even though MegaHAL has no negative numbers, its banking software gives exactly the same results as MicroHAL (which supports quaternions as its native format)! How come not using negative numbers changes nothing? Can it be that math exists in the real world regardless of the language we use to describe it?