Are Numbers Real?

[polymath257]

This is a topic I have thought about A LOT. I think we both invent and discover math. Let me see if I can explain.
Suppose I ask if the game of chess was invented or discovered. I think we all can agree it was invented. But, suppose I ask if, from a particular position in chess, there is mate in 4 moves. That is a question about some truth concerning those invented rules. And we can discover such truths even though the game of chess was itself invented.

Math is primarily an investigation of abstract formal systems. In such systems, we have axioms (basic assumptions). We choose those axioms, thereby inventing a topic in mathematics. Once those axioms have been chosen, however, we discover the consequences of those axioms.

So, that right triangles obey the Pythagorean equality is a discovery from the invented system of Euclidean geometry. If you choose other axioms, say those of non-Euclidean geometry, the Pythagorean equality would fail. There is then no 3-4-5 triangle.
The number pi can be defined in several very different ways, depending on the assumptions being made. In non-Euclidean geometry, though, it is no longer the ratio between the circumference and the diameter of a circle because there is no one such ratio, but many, depending on the size of the circle. The axiom system makes a difference in the truths. More clearly, truth depends on the assumptions made.

The same goes for numbers. We have some basic intuitions concerning numbers and such things as addition and multiplication. Those basic intuitions help us choose our axiom systems, thereby inventing a subject in mathematics. We can then discover whether certain results follow from those axioms. So, for example, in ordinary arithmetic, 13 is a prime number. But, if you use Gaussian integers, it is no longer prime. There is unique factorization into primes for ordinary arithmetic, but not if you look at certain algebraic number fields.

Each system of axioms has statements that it can prove, statements it can falsify (hopefully no overlap as that gives an inconsistency), and statements that it cannot decide. The latter class of statements can either be asserted or denied in conjunction with the other axioms and still have a system that is just as consistent. We get to choose in this case, based on our intuitions and our sense of aesthetics.

So, the answer to your question is that math is invented in that we choose our axioms. But after we do so, the consequences are discovered. Different systems will give different 'truths'.

Which system is best for describing the 'real world' is yet to be determined. That is a matter of experimentation and observation.

It seems that from your examples you are suggesting that mathematics is simply the analytical content of the axioms that we choose.  That seems fine as far as it goes, but in cases such as that of the Dirac quote below, we seem to somehow pack an extraordinary amount of analytical truth into a small number of axioms, from which such inferences as those that are made seems to correlate well with the real world almost in anticipation of real world truths.  There is nothing necessarily contradicting the possibility that we could choose such bountiful axioms so simply and easily, but it seems to bugger the imagination that we have done so purely by chance.  I know there is a good deal of fitting the axioms to the macroscopic reality of the world in mathematics today, but if you go back to, say, Euclid, and his basic postulates of geometry, so much that we can correlate at least to a reasonable approximation flows from that small set of axioms.  True, when you take things like the curvature of space-time into account, or possible non-Euclidean spaces, then derivations based upon those axioms will diverge somewhat, yet for our everyday macroscopic world, they seem more than coincidentally descriptive, and our ability to pack so much into such a small number of axioms seems almost magical.

It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.

~ P.A.M. Dirac

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Well, Euclid was the end result of a LOT of mathematical investigation. He took what had been 'applied math' used by the Egyptians and 'Babylonians' and combined it with some of the more theoretical material done by the Pythagoreans and Theatetus to figure out which axioms would do what he wanted to do. I would also point out that Euclid missed some assumptions that he nonetheless used implicitly concerning notions of betweenness.


So, no, the axioms are chosen, in almost every case, to describe some intuition that we have. At his point in time, even if the current axioms were found to be inconsistent (which Godel's results allow), there are certain mathematical statements what would survive into any new axiom system. For example, some version of the fundamental theorem of calculus will survive: it has simply been too useful for making models of physics. Many aspects of number theory would survive. But it is quite possible that large tracts of topology would go away.


So you are right, the axioms are NOT chosen by chance. They are chose specifically to abide by our intuitions and to be able to derive certain central results. Currently, all of mathematics is based on about a dozen axioms for set theory (the axioms of Zormelo and Fraenkl). Almost all the rest of mathematics can be expressed within those axioms (the exceptions are certain aspects of proper classes and extensions of the ZF axioms). These axioms were chosen and largely agreed to at the beginning of the last century.


And it *is* interesting that such a small number of axioms can be as expressive as they are. Again, this is partly why those axioms were chosen. The ability to model physical situations is another goal, although more often expressed as the ability to produce certain areas of math like differential geometry.


When applied to the 'real world', math becomes a language with unique expressive power. Because we get to define axioms however we want, we can adjust to any situation that we observe. This vastly increases the possibility of finding a model that works. Furthermore, it is often the case that mathematicians, in their curiosity, have already explored the systems that can later become useful for the physicists. This has lead to the claim that there is an 'unreasonable effectiveness to math'.
But this ignores the vast amounts of math that have *nothing* to do with the real world. Those that have little or no bearing on any question likely to arise in the real world. These areas of math are studied by mathematicians for their beauty, not for their applications. And those areas don't seem to give surprising new correlations with new physics.

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It's more that it is true because we say certain axioms are true. No, it is not a fact about all right triangles. As an example, there is a triangle on a sphere which has three right angles: pole to equator, 90 degrees along the equator, then back up to the pole. That is a perfectly legitimate triangle in spherical geometry. And the sum of its angles is 270 degrees, not 180. In such a triangle, even defining which is the hypothesis is problematic (it is a equilateral triangle after all). And the Pythagorean equality fails badly.

Thank you for your post. This sheds quite a bit of light on things. It is like Rob was saying before: math is only true because its underlying axioms are assumed to be true. But I don't think this presents a problem from mathematical Platonism or moral objectivism (not that I endorse mathematical Platonism or anything, just pointing this out).

What is an axiom? Let's define it as a basic assumption for the purposes of this discussion. I'm sure there is a more precise definition to be had among mathematicians, but that's pretty much what it means in philosophy and (I'm guessing) that's pretty much what it means in math. But a basic assumption can be correct, can it not?

I'm going a bit out of my wheelhouse here (and correct me if I'm wrong) but Euclidean geometry assumes that all its calculations transpire in flat 2D or 3D space. Those basic assumptions were so successful because, by and large, a great deal of the physical world conforms to those assumptions. My point (in the other thread) was that morality is an objective enterprise, just like math. In math, we didn't "choose" the axioms upon which any given system is based out of thin air. There was good reason for our assumptions... at the time when those systems were formulated, their assumptions were considered to be universal (of course they they aren't technically.... but pretty close).

The more modern viewpoint is that 'flat 2d and 3d space' are to be defined as those satisfying those axioms. They are very good approximations, yes. But once you start having to take the curvature of the Earth into account, it is better to use spherical geometry.

In regards to Jorm's post above, the success of mathematics seems to indicate that (at least some of) our basic assumptions (axioms) are correct. Otherwise, we got quite lucky in selecting them.

What I would disagree with concerning Rob is this:

It’s more like every single person draws up their own moral (mathematical) system, and so what is true in one system is not true in another. It just so happens that certain mathematical systems are so incredibly useful that it’s highly practical to all use the same one in most applied tasks.

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Maths applied through science can give us data and predict outcomes, but it can’t tell us which outcomes are preferable without also including exact criteria for what "preferable" means. It can’t do the ethics for you.

This seems to say

A) People are given a wide berth in selecting axioms, as if they can just pick whichever ones they want. I mean... they can... and the math will still work (I get that). But given the success of certain systems of mathematics in describing the physical world, this seems to suggest that we selected the "right" ones concerning those systems--ie. some of our basic assumptions were correct.  

B) That math is "useful" in physics in the same way that the myth of Santa Clause is useful in our ethical discourses with our children. That is, it's a useful story on a practical level, but it is otherwise made up. Rob seems to say, as mathematical fictionalists propose, math does not make truth statements. What is your take on Rob's post?

I would posit that (in ethics as well as math) a truth statement based upon certain axioms is a truth statement nonetheless... provided the basic assumptions (axioms) are correct. But I'm learning quite a bit here, so I'm going to stand back and listen a bit more before advancing any new claims.

I wouldn't say our assumptions were *correct* so much as they were *useful*. But they were originally selected *because* they were useful, so that isn't too surprising.

As for B). I see math as more of a language when it is applied than anything else. Because of how math is, it allows a great deal of flexibility in describing things. So, the axioms are generally built in such a way that we have a great deal of expressiveness in the system. They aren't so much assumptions as means of fitting things together to express what we observe.

But, and this is important (I think), there is a LOT of math that has nothing to do at all with the real world and modeling it. Nobody expects it to be relevant to and scientific investigation. It is studied only because mathematicians (such as we are) find the structures to be beautiful.