The evolution of logic, and the logic of evolution
The mathematician Eugene Wigner wrote a paper in 1960 asking what seems to me a very basic question - why does mathematics work in natural science? His answer was briefly that it often doesn't, and, as R. W. Hamming later observed in a follow-on paper, we see what we look for. Mathematics is a consequence, Hamming noted, of how we think.
He makes a remarkable comment. The evolution of man, he says, provided the model on which we have developed mathematics, quoting H. Mohr. This is set in the context of a rather arcane debate about whether mathematical concepts are Platonic ideals that always existed, or intuitions. It is, in effect, a third option - maths is something we evolved.
For a very long time, as both writers note, western thinkers followed Pythagoras and the later Greeks who believed that the universe was amenable to mathematical analysis, because the universe was fundamentally an ordered place that ran along mathematical rules. Our heads matched the way the world worked. There have been some more absurd versions of this - Plato's belief that the world was round because it was like the head, and what was true of the head must be true of the world is an extreme version of this, but it has been, for 2500 years, the default assumptions.
In the late 19thC and early 20thC, thinkers like Gottlob Frege and Bertrand Russell held a view sometimes known as "logicism" - that mathematics reduces to logic. Under this view, the reason why maths works is that logic works, and the world is fundamentally logical. But what are these logical truths on which we build our mathematical edifices? Plato's account, as exemplified by the slave boy he tricks into working out a mathematical truth by asking leading questions, is that we remember them from before we were born, when we, or at any rate our souls, existed among the eternal forms.
But the way Plato gets this result is in itself a hint about what is really going on. Maths is a language, and what can be expressed in it, is what we need to express, based on how we live; what Wittgenstein called out "forms of life", both biological and cultural. Hamming assumes that our biology is paramount - but I think he overestimates the biological implications of logic. And so we come to the book that inspires this little essay:
William S. Cooper, in a book entitled The Evolution of Reason: Logic as a Branch of Biology (Cambridge University Press: Cambridge UK, 2001, ISBN 0-521-79196-0), makes an even more remarkable claim - maths may reduce to logic, but there are a number of logics, and at base is decision theory, which is reducible to "life-history strategy theory", which is reducible to evolutionary theory.
Logic works, in short, because it is an abstraction out of the principles of the way evolution works.
This turns the entire way we think about logic and maths on its head. It is, in effect, the exact opposite and reverse of mathematical platonism. It replaces, shoulders aside, really, intuitionism. It is remarkable. Cooper claims that logic is a branch of biology.
It's not the first such book to make a claim about what had been thought of as abstract categories being biological. Ruth Garrett Millikan argued that thought and language, and in particular ideas like functions, were biological rather than platonic. But logic...
Willard Van Ormand Quine, the great American philosopher, a pragmatist and to a great degree a nominalist, held that "Creatures inveterately wrong in their inductions have a pathetic, but praiseworthy, tendency to die before reproducing their kind" (in From a Logical Point of View, Harvard University Press, Cambridge MA, 1953). Of course, it's only praiseworthy if you happen to think that the greatest good is to produce good inductive beings, and there is surely more to it than that, but the point is valid. Logic works not because it has been brought to us on Moses' Stone Tablets, as Hamming observes, but because we use the logic that works.
So, what is the logic of evolutionary theory? It rests, says Cooper, in the decision trees that organisms employ by implication in adapting to their world. Logic is the logic of survival and selection. We have seen many attempts to make use of Darwin's remarkable idea of natural selection. But like all these accounts, one thing that Cooper appears to overlook is the lack of selection in a good deal of evolution. Much evolution is in fact not the signal of selection, but the noise of mutation, drift and contingency.
In subsequent posts, I shall explore some of Cooper's ideas further. There will be no equations, because I am amathematical in my skill set, so do not fear. But there are some deep issues here I would like to poke at with a suitably long stick. Comments and suggestions are, of course, welcomed.
He makes a remarkable comment. The evolution of man, he says, provided the model on which we have developed mathematics, quoting H. Mohr. This is set in the context of a rather arcane debate about whether mathematical concepts are Platonic ideals that always existed, or intuitions. It is, in effect, a third option - maths is something we evolved.
For a very long time, as both writers note, western thinkers followed Pythagoras and the later Greeks who believed that the universe was amenable to mathematical analysis, because the universe was fundamentally an ordered place that ran along mathematical rules. Our heads matched the way the world worked. There have been some more absurd versions of this - Plato's belief that the world was round because it was like the head, and what was true of the head must be true of the world is an extreme version of this, but it has been, for 2500 years, the default assumptions.
In the late 19thC and early 20thC, thinkers like Gottlob Frege and Bertrand Russell held a view sometimes known as "logicism" - that mathematics reduces to logic. Under this view, the reason why maths works is that logic works, and the world is fundamentally logical. But what are these logical truths on which we build our mathematical edifices? Plato's account, as exemplified by the slave boy he tricks into working out a mathematical truth by asking leading questions, is that we remember them from before we were born, when we, or at any rate our souls, existed among the eternal forms.
But the way Plato gets this result is in itself a hint about what is really going on. Maths is a language, and what can be expressed in it, is what we need to express, based on how we live; what Wittgenstein called out "forms of life", both biological and cultural. Hamming assumes that our biology is paramount - but I think he overestimates the biological implications of logic. And so we come to the book that inspires this little essay:
William S. Cooper, in a book entitled The Evolution of Reason: Logic as a Branch of Biology (Cambridge University Press: Cambridge UK, 2001, ISBN 0-521-79196-0), makes an even more remarkable claim - maths may reduce to logic, but there are a number of logics, and at base is decision theory, which is reducible to "life-history strategy theory", which is reducible to evolutionary theory.
Logic works, in short, because it is an abstraction out of the principles of the way evolution works.
This turns the entire way we think about logic and maths on its head. It is, in effect, the exact opposite and reverse of mathematical platonism. It replaces, shoulders aside, really, intuitionism. It is remarkable. Cooper claims that logic is a branch of biology.
It's not the first such book to make a claim about what had been thought of as abstract categories being biological. Ruth Garrett Millikan argued that thought and language, and in particular ideas like functions, were biological rather than platonic. But logic...
Willard Van Ormand Quine, the great American philosopher, a pragmatist and to a great degree a nominalist, held that "Creatures inveterately wrong in their inductions have a pathetic, but praiseworthy, tendency to die before reproducing their kind" (in From a Logical Point of View, Harvard University Press, Cambridge MA, 1953). Of course, it's only praiseworthy if you happen to think that the greatest good is to produce good inductive beings, and there is surely more to it than that, but the point is valid. Logic works not because it has been brought to us on Moses' Stone Tablets, as Hamming observes, but because we use the logic that works.
So, what is the logic of evolutionary theory? It rests, says Cooper, in the decision trees that organisms employ by implication in adapting to their world. Logic is the logic of survival and selection. We have seen many attempts to make use of Darwin's remarkable idea of natural selection. But like all these accounts, one thing that Cooper appears to overlook is the lack of selection in a good deal of evolution. Much evolution is in fact not the signal of selection, but the noise of mutation, drift and contingency.
In subsequent posts, I shall explore some of Cooper's ideas further. There will be no equations, because I am amathematical in my skill set, so do not fear. But there are some deep issues here I would like to poke at with a suitably long stick. Comments and suggestions are, of course, welcomed.
posted by John S. Wilkins @ 4:42 pm
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