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I'd like to talk about a couple of simple thought experiments invented by the mathematician and philosopher Gottlob Frege, which for me really nail the relationship between mathematics and the physical world, and at the same time raise some fundamental questions in the philosophy of mathematics.
Suppose you have a vessel containing 2cc of liquid. To this, you add a further 3cc of liquid. How much is in the vessel now? Well, 2+3 = 5, so obviously it's 5cc. But the liquid you added wasn't the same as the liquid that was in there: the two underwent a chemical reaction, emitting some gas, and so the volume of the liquid in the vessel is actually less than 5cc. Or suppose you have five things, and you then add two more half-things. You've got six things, right? Except the "things" were pairs of boots, and the half-things were individual boots, but both individual boots were left boots. You have twelve boots, but only five pairs.
In both cases, you have some physical system (liquid in a vessel, pairs of boots in a rack) that you're trying to model using some mathematical formalism (whole numbers and addition, fractions and addition). In each case, it turns out that the model isn't a very good one, as it incorrectly predicts the behaviour of the system. But this doesn't mean that 2+3 is not equal to 5, or that 5 + 1/2 + 1/2 is not equal to 6! Nor does it mean that there's no mathematical model that would fit - in both cases, it would be pretty easy to construct one. It just means that we've chosen the wrong models. Standard arithmetic still works fine, it just happens to be the wrong thing in this case.
This is how science works: we take the phenomena we're interested in, and construct models to describe them. We then deduce consequences of these models, and test to see if they're true. If the consequences don't show up in the real system under consideration, we know that something was wrong with our original assumptions. Frege's thought experiments are so enlightening because the mathematics involved is trivial, allowing us to concentrate on the modelling. In good cases, we can construct precise models that fit the system well, and which tie in to well-known mathematics (which is by definition the study of the consequences which flow from precisely-defined assumptions). In bad cases, it's hard to find rules that work well, the rules you have are rules of thumb following no obvious pattern, and they don't fit into any established mathematical theory. Biology suffers from all three problems, which is why mathematical biology is such an exciting area: hard problems are interesting.
With this perspective, the often-claimed "unreasonable effectiveness of mathematics in the natural sciences" becomes entirely reasonable: if we can find good models, then mathematics will be extremely effective. Mathematics is the deduction of consequences from precise axioms: if those axioms are good fits to the real systems under consideration, then our deductions will be too. If we can't find good models, then mathematics will be much less useful: at the time Wigner made his comment about "unreasonable effectiveness", science was dominated by the study of situations where we did have good models. Now, science has expanded outwards to hit the gaps in our mathematical knowledge, and this is much clearer. Eric Raymond has written an excellent article on this stuff, concentrating on the feedback loop between mathematics and science. My favourite quotation about this is definitely Bertrand Russell's, though:
Here's how Frege's experiments tie into philosophy: there's a certain view of mathematics (advanced by John Stuart Mill, among others) that says that a mathematical statement is true if and only if it's true for every real-world instantiation. So 2 + 3 = 5 if and only if every time you have two things and add three things you get five things. Frege was marvellously dismissive of this approach1, describing it as a "gingerbread and pebble arithmetic", and his examples show that the failure of two things and three things to give five things does not mean that 2 + 3 != 5; you must satisfy all the assumptions of the model to invalidate it like that. But in general, there is no physical system which satisfies all the assumptions of a given mathematical model perfectly: for instance, there are no perfect circles anywhere in the Universe. More simply, the number of particles in the Universe is around 1080: if Mill were right, what meaning could we assign to numbers larger than that? It's this kind of problem that drives mathematicians into the arms of Platonism, the idea that there exists a realm of "ideals", apart from the physical Universe, which contains actual instances of every mathematical object. Of course, this raises just as many difficulties as it solves...
Frege's story's rather a sad one, as it happens. He did pioneering work in the philosophy of language and in logic (he's got a solid claim to be one of the three great logicians of all time, along with Boole and Aristotle, and without his work much of modern mathematics would be literally unthinkable). In 1903 the culmination of his life's work, the Basic Laws of Arithmetic (Grundgesetze der Arithmetik), was on the verge of publication: in it, he claimed to show that arithmetic (and thus all of mathematics) could be derived from the obvious truths of logic. As it was about to go to press, he received a letter from Bertrand Russell informing him of what is now known as Russell's Paradox. This blew the entire enterprise out of the water. Frege inserted a preface to the effect of "This doesn't work any more, but I hope you find it interesting", then went off and had a nervous breakdown.
1 If you can get hold of a copy of his (sadly out-of-print) Foundations of Arithmetic, I strongly recommend it, if only for the chapter contra Mill.
Suppose you have a vessel containing 2cc of liquid. To this, you add a further 3cc of liquid. How much is in the vessel now? Well, 2+3 = 5, so obviously it's 5cc. But the liquid you added wasn't the same as the liquid that was in there: the two underwent a chemical reaction, emitting some gas, and so the volume of the liquid in the vessel is actually less than 5cc. Or suppose you have five things, and you then add two more half-things. You've got six things, right? Except the "things" were pairs of boots, and the half-things were individual boots, but both individual boots were left boots. You have twelve boots, but only five pairs.
In both cases, you have some physical system (liquid in a vessel, pairs of boots in a rack) that you're trying to model using some mathematical formalism (whole numbers and addition, fractions and addition). In each case, it turns out that the model isn't a very good one, as it incorrectly predicts the behaviour of the system. But this doesn't mean that 2+3 is not equal to 5, or that 5 + 1/2 + 1/2 is not equal to 6! Nor does it mean that there's no mathematical model that would fit - in both cases, it would be pretty easy to construct one. It just means that we've chosen the wrong models. Standard arithmetic still works fine, it just happens to be the wrong thing in this case.
This is how science works: we take the phenomena we're interested in, and construct models to describe them. We then deduce consequences of these models, and test to see if they're true. If the consequences don't show up in the real system under consideration, we know that something was wrong with our original assumptions. Frege's thought experiments are so enlightening because the mathematics involved is trivial, allowing us to concentrate on the modelling. In good cases, we can construct precise models that fit the system well, and which tie in to well-known mathematics (which is by definition the study of the consequences which flow from precisely-defined assumptions). In bad cases, it's hard to find rules that work well, the rules you have are rules of thumb following no obvious pattern, and they don't fit into any established mathematical theory. Biology suffers from all three problems, which is why mathematical biology is such an exciting area: hard problems are interesting.
With this perspective, the often-claimed "unreasonable effectiveness of mathematics in the natural sciences" becomes entirely reasonable: if we can find good models, then mathematics will be extremely effective. Mathematics is the deduction of consequences from precise axioms: if those axioms are good fits to the real systems under consideration, then our deductions will be too. If we can't find good models, then mathematics will be much less useful: at the time Wigner made his comment about "unreasonable effectiveness", science was dominated by the study of situations where we did have good models. Now, science has expanded outwards to hit the gaps in our mathematical knowledge, and this is much clearer. Eric Raymond has written an excellent article on this stuff, concentrating on the feedback loop between mathematics and science. My favourite quotation about this is definitely Bertrand Russell's, though:
It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance.
Here's how Frege's experiments tie into philosophy: there's a certain view of mathematics (advanced by John Stuart Mill, among others) that says that a mathematical statement is true if and only if it's true for every real-world instantiation. So 2 + 3 = 5 if and only if every time you have two things and add three things you get five things. Frege was marvellously dismissive of this approach1, describing it as a "gingerbread and pebble arithmetic", and his examples show that the failure of two things and three things to give five things does not mean that 2 + 3 != 5; you must satisfy all the assumptions of the model to invalidate it like that. But in general, there is no physical system which satisfies all the assumptions of a given mathematical model perfectly: for instance, there are no perfect circles anywhere in the Universe. More simply, the number of particles in the Universe is around 1080: if Mill were right, what meaning could we assign to numbers larger than that? It's this kind of problem that drives mathematicians into the arms of Platonism, the idea that there exists a realm of "ideals", apart from the physical Universe, which contains actual instances of every mathematical object. Of course, this raises just as many difficulties as it solves...
Frege's story's rather a sad one, as it happens. He did pioneering work in the philosophy of language and in logic (he's got a solid claim to be one of the three great logicians of all time, along with Boole and Aristotle, and without his work much of modern mathematics would be literally unthinkable). In 1903 the culmination of his life's work, the Basic Laws of Arithmetic (Grundgesetze der Arithmetik), was on the verge of publication: in it, he claimed to show that arithmetic (and thus all of mathematics) could be derived from the obvious truths of logic. As it was about to go to press, he received a letter from Bertrand Russell informing him of what is now known as Russell's Paradox. This blew the entire enterprise out of the water. Frege inserted a preface to the effect of "This doesn't work any more, but I hope you find it interesting", then went off and had a nervous breakdown.
1 If you can get hold of a copy of his (sadly out-of-print) Foundations of Arithmetic, I strongly recommend it, if only for the chapter contra Mill.
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I think I walk pass his house occasionally.
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Have you read any of his books? I've only read the Grundlagen (in translation, at that: makes it a bit tricky when he spends a chapter dissecting the significance of the word "Einheit", which as far as I can tell has no precise equivalent in English). The mathematics isn't so great - the holes are pretty obvious, post-Russell - but the style is wonderful, and I found the philosophical parts pretty mind-expanding. I'd like to read the Begriffsschrift some day...
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If x + y -> z
Volume(x) + Volume(y) != Volume(z)
but: Mass(x) + Mass(y) = Mass(z); N(x) + N(y) = N(z)
Of course in contemporary physics we've found that in general mass and number of atoms are not convserved (for instance in a radioactive decay), however there are still quantities which are conserved, the ones which most people will be most familiar with being energy, linear momentum and angular momentum.
Of course all that is fairly well known - you were probably taught it in A-level maths and physics. However there is a beautiful theorem known as Norther's theorem, which states that for every continuous symmetry there is a conserved quantity; e.g.
If we have a spatialy symmetric situation (i.e. the laws of physics are invarient between two different points, x and y) there will be a quantity (linear momentum) which is conserved between x and y.
If we have a rotational symmetric situation (i.e. the laws of physics are invarient between two angles, q1 and q2) there will be a quantity (angular momentum) which is conserved between points q1 and q2.
If we have a time symmetric situation (i.e. the laws of physics are invarient between two times, t1 and t2) then there will be a quantity (energy) which is conserved between times t1 and t2.
This really blows my mind - taking fairly simple and opaque observations about the Universe (the conservation laws) implies something very deep about its structure (that it has a symmetry in space and time). Its one of those great examples where mathematical implications has said something really significant about physics.
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Categorisation: well, groups are exactly one-object categories in which every arrow is invertible, so I have no problem with that terminology :-)