There's this age old question of where mathematics come from: is it a product of the human mind, or does it exist independently of it? Here's the labels:

Realism: A realist believes that math exists outside of our minds. We discover mathematics, we don't invent it. Mathematics is real.

Anti-realism: An anti-realist believes that mathematics exists within our minds. We invent mathematics, we don't discover it. Mathematics is not physically real.

Realists argue that mathematicians make discoveries long before they become useful in physics, and that can't be a coincidence. When Einstein wanted to develop general relativity, non-Euclidean geometry had already been developed. Quantum mechanics is based on Hilbert spaces (functional analysis), which had also already been developed by mathematicians.

Anti-realists, on the other hand, argue that the truth of mathematical statements depend on assumptions, i.e. mathematics does not have any unconditional truths (apart from logical tautologies, such as "A implies A"). For example, let's reconsider geometry: there are various kinds of them, the Euclidean, and many non-Euclidean. For a long time the ancients saw Euclidean geometry as self evident, although we today know that this "self evident fact" is incorrect.

But let's take one question very seriously: why are the properties of the universe so mathematical?

It's almost as if the universe is made of mathematics, given that the laws of physics are mathematical relations (and moreover, they may all be computable). This hints towards a possible physical answer to the great debate between these two camps.

The universe may appear to be mathematical because it IS mathematical. The universe is a formal axiomatic system, and we exist as part of those axioms. The assertion may in fact be physically provable: if we discover a theory of everything (i.e. the ultimate laws of physics), then the universe is obviously mathematical, and realism turns out to be true: pi is not in the sky, but in the formalization of the laws of nature. As a special case, we it turns out that all physical processes of nature are computable, then the universe is not only mathematical, but a Turing-equivalent computer.

On the other hand, it may turn out that the universe is not mathematical, and we "discover" mathematical properties of the universe only because our mind creates them. This may sound crazy at first, but one may note that many concept are in fact purely man made (although indispensable to everyday life): concepts such as human, car, alive/dead, and temperature all rely on human interpretation of a huge amount of matter ("a macroscopic phenomena"). It is not fundamental to the universe: a car is just an arrangement of particles/energy ("a state"), no more special than a pile of junk. Put differently, the universe may be "blind" to our high level concepts, and that may actually include most of physics, possibly with the exception of quantum mechanics (although that is not certain either). From this perspective, the universe is not mathematical: we interpret it as having mathematical properties, and that approximate some things very well, but will never grasp the full extent of it (that is the case right now). In that case, anti-realists win: mathematics is only in the mind (yet very useful to our existence).

Realism: A realist believes that math exists outside of our minds. We discover mathematics, we don't invent it. Mathematics is real.

Anti-realism: An anti-realist believes that mathematics exists within our minds. We invent mathematics, we don't discover it. Mathematics is not physically real.

Realists argue that mathematicians make discoveries long before they become useful in physics, and that can't be a coincidence. When Einstein wanted to develop general relativity, non-Euclidean geometry had already been developed. Quantum mechanics is based on Hilbert spaces (functional analysis), which had also already been developed by mathematicians.

Anti-realists, on the other hand, argue that the truth of mathematical statements depend on assumptions, i.e. mathematics does not have any unconditional truths (apart from logical tautologies, such as "A implies A"). For example, let's reconsider geometry: there are various kinds of them, the Euclidean, and many non-Euclidean. For a long time the ancients saw Euclidean geometry as self evident, although we today know that this "self evident fact" is incorrect.

But let's take one question very seriously: why are the properties of the universe so mathematical?

It's almost as if the universe is made of mathematics, given that the laws of physics are mathematical relations (and moreover, they may all be computable). This hints towards a possible physical answer to the great debate between these two camps.

The universe may appear to be mathematical because it IS mathematical. The universe is a formal axiomatic system, and we exist as part of those axioms. The assertion may in fact be physically provable: if we discover a theory of everything (i.e. the ultimate laws of physics), then the universe is obviously mathematical, and realism turns out to be true: pi is not in the sky, but in the formalization of the laws of nature. As a special case, we it turns out that all physical processes of nature are computable, then the universe is not only mathematical, but a Turing-equivalent computer.

On the other hand, it may turn out that the universe is not mathematical, and we "discover" mathematical properties of the universe only because our mind creates them. This may sound crazy at first, but one may note that many concept are in fact purely man made (although indispensable to everyday life): concepts such as human, car, alive/dead, and temperature all rely on human interpretation of a huge amount of matter ("a macroscopic phenomena"). It is not fundamental to the universe: a car is just an arrangement of particles/energy ("a state"), no more special than a pile of junk. Put differently, the universe may be "blind" to our high level concepts, and that may actually include most of physics, possibly with the exception of quantum mechanics (although that is not certain either). From this perspective, the universe is not mathematical: we interpret it as having mathematical properties, and that approximate some things very well, but will never grasp the full extent of it (that is the case right now). In that case, anti-realists win: mathematics is only in the mind (yet very useful to our existence).

I think the question if our universe "really" is mathematical structure is a nonsensical question, because we'll never be able to prove or disprove it.

ReplyDeleteIf we find a theory of everything (i.e. laws of physics from which every observed phenomena can be explained) then it will provide evidence that it is a mathematical structure.

DeleteSure, it could be that we're wrong, it just looks like the "ultimate" laws, and after decades or centuries turns out to be an over simplification. But if that makes it "nonsensical", then science itself is nonsensical, since this is precisely what the scientific method does (and exactly what happened to classical physics).

Science is about making observations and finding the shortest description of all those observations. I think we probably will find a nice ToE. Possibly even a perfectly discrete and deterministic one.

DeleteBut the question is if our universe really "is" this mathematical structure Or if our universe just emerges from a deeper underlying non-mathematical reality. Things which are not describable by math are not understandable at all. So it shouldn't even make sense to talk about such things.

For math anti-realists the word "math" often means only the math which you can see in books or which will likely be written down in the future.

For math realists the word "math" usually means every possible form math which you could in principle think of. So for them math is equivalent to everything you could ever imagine. And of course frequent usage of the word "exists" by mathematicians reinforces the intuition of mathematical realism. That I think is why most mathematicians are math realists.

What is more appropriate to say?:

That Benoit Mandelbrot constructed the Mandelbrot set

or that Benoit Mandelbrot discovered the Mandelbrot set?

I (Gabriel Leuenberger) would choose the second one.

You have perfectly understood the issue, so even if you don't believe in anti-realism (and the point of my post is not to take sides, but to illuminate the relationship between mathematical philosophy and physical reality), you have made sense of the distinction, yet you call it "nonsensical" :)

DeleteYour objections in terms of the scientific method as a means to obtaining a shortest description is valid; yet the question does make sense as it is not too hard to actually observe a non-mathematical universe. Some examples:

1. The laws of physics are not fixed, they sometimes change (compare with False Vaccum in physics).

2. There is an infinite regress of physical laws. We peel them off layer by layer, but there is no end to the process.

3. The universe is a formal axiomatic system with non recursively enumerable axioms.

(1) would probably mean the end of every intelligent form, so perhaps the question is not that interesting.

(2) is a good example: the universe does have structure, but infinitely many layers of it depending on how far you "zoom" in. It's just that we can't reach all the way down. But why wouldn't we be able to capture the pattern of change in the laws from layer to layer? Presumably because of (3): the pattern is non-computable (I'm assuming hypercomputation is impossible).

In (3) , we capture some parts of reality, but since the axioms are non-computable, a ToE would have to have infinite description length (and we only have finitely much memory to work with).

Ok it makes more sense now. Science would have to peel of a Lot of layers until we can assume that there's no end to the process.

DeleteWhich is off course unlikely to happen because we'd probably encounter some almost impenetrable layer.