One particularly contentious question in the foundations of both mathematics and physics is whether mathematics is discovered or invented. Related to this is whether mathematics is the way the world actually is or if it is simply a way in which we can model the world.
This is a particularly difficult question to answer since it is quite clear that there are physically impossible situations that can be spoken about in mathematics. The best example I can think of relates directly to some things I bitched about on this blog in recent weeks concerning Brikhoff’s theorem (which I won’t rehash in the interest of brevity): taking n copies of a quantum channel should start to approach something that can be approximated by unitaries as n approaches infinity, i.e. in the asymptotic limit, n copies of a channel have a unitary representation (roughly speaking and if the theorem is correct). But, from a physical point of view, this is ridiculous since it is literally impossible for an infinite number of channels (or anything else, for that matter) to physically exist. Yet whole branches of mathematics are devoted to working with infinity (infinite sets, for example).
Clearly some mathematical results have been originally thought to be entirely abstract only to much later find some application in the physical world. But it could be argued that concrete, mathematical analyses of infinite things could never, by their very definition, find an application in the physical world. Does this mean that mathematics is inherently discovered then? Or could it be taken as “evidence” for the Many-Worlds Interpretation (MWI), i.e. the only way to have a physically infinite result is to have a physically infinite number of universes?
Quantum Moxie 
I like the title of this post as it is also the title of a course I am teaching this semester. I have a couple comments in regards to applications of infinite mathematics. First, we would not have a rigorous definition of derivative or integral without limits to infinity. That example is very concrete. Now let me try for something more abstract. We are limited to finite observations. However, this does not imply that what we are observing is not infinite. Back to the concrete and more closely related to the asymptotic Birkhoff result. It is very common to replace (for practical reasons) an asymptotic result with the less formal “if we have a enough”. Here I am thinking of something like the Central Limit Theorem in Statistics. No finite sample is necessarily normal, but a large enough sample is approximately normal.
Steve,
I actually talked about derivatives in quantum mechanics today. On the one hand, one could argue that the physicality of a derivative is limited by the uncertainty principle, i.e. instantaneous changes are physically unattainable, they just appear to be from our macroscopic (coarse-grained, as Brukner puts it) point-of-view. On the other hand, one could also argue (well, maybe only one as insane as me), that the fact that we can work with infinities in mathematics is evidence of the many-worlds interpretation of QM (which I’m agnostic on, by the way) since you’d need an infinite number of universes in order to see an infinite realization of something.
> Related to this is whether mathematics is the way the world actually is or if it is simply a way in which we can model the world.
It’s pretty clear that (2) is true, we use mathematics to successfully model the world all the time. I personally think (1) is correct but that the search for the mathematical model that REALLY underlies everything is not going to successfully conclude any time soon.
Geordie,
That’s a good point. It clearly does model the world (I’m not aware of any physical situations that mathematics doesn’t model, but would be curious to know if there are any out there). So the question is really whether it represents the world, i.e. is the language of nature, or not. Or, perhaps, we simply haven’t discovered “math” as a whole, but rather various approximations of it.
I think what we haven’t discovered (and maybe can’t) is “the way things really are”. I suspect that if we ever discover this, mathematics as we currently understand it will be sufficient to model it.
Interesting. I suspect you may be right about the fact that we might not be able to discover how things really are. You’re also probably right that mathematics as we currently understand it should be sufficient to model it since mathematics is rooted in logic and, if math weren’t sufficient, then the universe would have, at its root, a different logic than the one we know which doesn’t make any sense.
Always an interesting question. Zeno’s paradox describes a real physical situation that requires the concept of infinity to solve, unless the world is discrete at the Planck scale, in which case there’s just a large, albeit finite, number of steps from A to B. If you’re going to invoke MWI then a logical, well thought out statement of how it would apply would be preferred to just throwing it out there.
Joe,
This is a blog. Furthermore, this is my blog. Thus I am invoking my right as the blogger to randomly throw stuff out there. It’s called “thinking out loud” and sometimes doing so yields interesting feedback from anyone who happens to be “listening.” 😉
Gregory Chaitin in a 2006 artice in Scientific American says:
Yet both fields are similar. In physics,
and indeed in science generally, scientists
compress their experimental observations
into scientific laws. They then
show how their observations can be deduced
from these laws. In mathematics,
too, something like this happens—
mathematicians compress their computational
experiments into mathematical
axioms, and they then show how to deduce
theorems from these axioms.
If Hilbert had been right, mathematics
would be a closed system, without
room for new ideas. There would be a
static, closed theory of everything for
all of mathematics, and this would be
like a dictatorship. In fact, for mathematics
to progress you actually need
new ideas and plenty of room for creativity.
It does not suffice to grind away,
mechanically deducing all the possible
consequences of a fixed number of basic
principles. I much prefer an open system.
I do not like rigid, authoritarian
ways of thinking.
Another person who thought mathematics
is like physics was Imre Lakatos,
who left Hungary in 1956 and later
worked on philosophy of science in England.
There Lakatos came up with a
great word, “quasi-empirical,” which
means that even though there are no
true experiments that can be carried out
in mathematics, something similar does
take place. For example, the Goldbach
conjecture states that any even number
greater than 2 can be expressed as the
sum of two prime numbers. This conjecture
was arrived at experimentally,
by noting empirically that it was true for
every even number that anyone cared to
examine. The conjecture has not yet
been proved, but it has been verified up
to 1014.
I think that mathematics is quasiempirical.
In other words, I feel that
mathematics is different from physics
(which is truly empirical) but perhaps
not as different as most people think.
I have lived in the worlds of both
mathematics and physics, and I never
thought there was such a big difference
between these two fields. It is a matter
of degree, of emphasis, not an absolute
difference. After all, mathematics and
physics coevolved. Mathematicians
should not isolate themselves. They
should not cut themselves off from rich
sources of new ideas.
“If Hilbert had been right, mathematics
would be a closed system, without
room for new ideas. There would be a
static, closed theory of everything for
all of mathematics, and this would be
like a dictatorship. In fact, for mathematics
to progress you actually need
new ideas and plenty of room for creativity.”
I believe mathematics is discovered and not created. The truths are static. What changes is how many truths we have discovered. Hilbert put together a list of open problems in 1900(not all of which have been solved). So whatever his views were, they certainly left open plenty of room for progress.
I do agree that conjectures in mathematics come from empirical data. However, mathematicians require more than overwhelming empirical data to convert a conjecture to a theorem.
Also, I am quite sure Goldbach’s conjecture has been verified beyond 1014. Should that read 10^14?
Robert,
Those are some very interesting thoughts. I too have inhabited both worlds (math and physics). I had always thought they were very similar (the fact that advances in mathematics have come out of physics is somewhat surprising to some people). But then I got wrapped up in a debate about infinity and I began to realize they were more different than I had originally thought. That isn’t to say they are all that different, just more so than I had ever thought. But I don’t see it as a bad thing either. What I actually see it as is an opportunity. I am interested in those places where mathematics just begins to diverge from the physical world. I think we might learn a lot about both mathematics and physics this way.
To get from the blackboard to the door, you must first go to the midpoint of the two objects. Once at the midpoint between the door and board, you must go to the midpoint between the door and the previous midpoint. In fact, you must hit infinitely many midpoints if you’re ever going to get to out of the room. How can you complete an infinite amount of tasks in finite time? Since, I have yet to get trapped in a classroom, it seems as though we realize infinity more than we thought. (More rigor: The distance from the board to the door can be represented as an infinite series)
In response to Steve S. who says: “you must hit infinitely many midpoints if you’re ever going to get to out of the room. How can you complete an infinite amount of tasks in finite time?”
Assuming a constant velocity (which is useful for simplicity, but not logically necessary), the travel time to each midpoint is half of the previous travel time. In other words, as you divide the distance into thinner and thinner slices, you are also dividing the associated time interval into thinner and thinner slices.
More formally: some infinite series converge to a single definite number. “1/2 + 1/4 + 1/8 + 1/16 + …” is an example of a convergent geometric series that in fact is equal to 1.
Aristotle correctly pointed out that there is a big difference between ‘infinite in divisibility’ and ‘infinite in extent’.
are far as whether mathematics is created or really exists is a question that will most probabily never answered because to truly answer that question one must intuitivly understand everything in math which under godels incompletness theorem is impossible.the question i put to u is what lies beyond the axiom?which like mathematics is beyond the human min