FOM: Evolution and reason wtait at
Thu Apr 2 16:33:51 EST 1998

I want to begin to respond, with thanks for their comments on my posting 
about history, to Randall Holmes, Joe Shipman, Walter Felscher, Steve 
Simpson and Neil Tennant. Let me begin with Neil.

>Bill Tait has invoked evolutionary theory as a possible
>explanation (validation?) of our capacity for logical and mathematical

The context of my remarks was the issue of the `unreasonable 
effectiveness of ...'; so I was invoking natural selection as a possible 
explanation of *this*---which is quite different from either an 
explanation or a validation of our capacity for .... , or at least would 
be on quite reasonable interpretations of the latter.

>But if the individual's a priori is the species' a posteriori, this raises
>the question of what it is *in the physical world* that is exerting the
>selective force(s) in such a way as to produce brains that operate (at the
>level of thought) in accordance with those so-called a priori's.

The term `a priori' originally referred to propositions which follow 
`from what is prior', i.e. from first principles---say axioms. So it 
concerned propositions in deductive sciences such as geometry or, more 
generally in classical Greece, in the exact sciences. Although it has 
been used in other ways, this is the only sense of `a priori' that I 
really understand; and so I don't understand `the individual's a priori' 
nor the `species' a posteriori'. Note that `a priori' in the original 
sense presupposes logic, at least in the sense of what counts as a valid 

The question ``What is it *in the physical world* that is exerting the 
selective force(s) in such a way to produce X?'', whatever X is, seems 
very far from the ideology of Darwin's theory, since it seems to be 
asking for a teleological force. I may be misunderstanding you; but I 
can't find a meaning for the question that makes sense in the context of 
evolution *by natural selection*.

>Bill, would you accept the following Uniformity Postulate for Logic?:
>	There is a system S of logic such that:
>	whatever the laws of physics, chemistry etc. might be in any
>	possible world in which reasoning creatures could evolve, their
>	reasoning would have to obey the laws of S.

I am weak on both the subject of possible worlds and that of rational 
creatures. For example, in this world I am not sure whether chimps who 
figure out a method of using levers to get food, a method that they then 
can repeat, are rational. Where does one draw the line? On using symbols 
in the reasoning? Is it clear that they don't? On having language in some 
stricter sense? Where do we draw *that* line?

I don't think that ``their reasoning would have to obey the laws S'' is 
entirely clear. I am assuming that you are refering to the norms for 
correct reasoning that they (or at least the official reasoners among 
them) hold to; but even so understood, in order for it to be meaningful, 
there would have to be a translation of at least certain of their 
expressions into English, say, as logical constants. But a big part of 
the evidence for such a proposed translation being correct would be that, 
on that proposed translation, they obeyed our laws of logic? Isn't there 
a circle?

>Would you also accept the following analogue for mathematics?:
>	There is a system T of theorizing about numbers such that:
>	whatever the laws of physics, chemistry etc. might be in any
>	possible world in which reasoning creatures could evolve, their
>	theorizing would have to be consistent with T (i.e.
>	extendible so as to include T, upon suitable translation).

By ``a system T of theorizing about numbers'' do you mean simply an 
axiomatic theory of numbers?  Any two consistent theories in our sense, 
say first-order theories, are consistent with one another upon suitable 
translation: just translate the non-logical vocabulary of one into a 
vocabulary disjoint from that of the other. It is not clear what 
constraints you want to put on translations to make your principle 

>I want now to pose a dilemma. If you accept these uniformity postulates,
>then all that the evolutionary story delivers is an account of how we
>attained to S and T, not why S and T are (necessarily) valid/true.

I wanted the theory of evolution by natural selection to account for the 
*effectiveness* of mathematics in explaining, etc, natural phenomena: 
that is where my posting began. Since I am not clear about what exactly S 
and T are, I am not sure what you mean by their truth or validity. But 
let me say that neither term, ``truth'' or ``validity'', is a term that I 
would apply to mathematics or fragments of it such as number theory, as a 
whole. I understand what I would mean by a mathematical---say an 
arithmetical--- proposition being true: it means that it is provable, 
where `provable' is an open-ended notion allowing for our admitting 
stronger and stronger methods of proof. [Of course this open-endedness is 
no problem for mathematics because *this* notion of truth (as opposed to 
that of the truth of a formal sentence in a model) plays no role in 
mathematics.] But the notion of arithmetic as a whole, e.g. as it has 
developed, being true or false has no meaning for me. (Of course, if it 
should turn out that god deceived us and it is inconsistent, I might want 
to say it is false. More likely, though, I would turn my face to the wall 
and suck my thumb.)

I want `the evolutionary story' to account for how creatures like us, 
with our propensity ``to seek in theories (logoi) the truth of the things 
that are'', developed. 

>I prefer the first horn of the dilemma, and so am prepared to look elsewhere
>than evolutionary theory for the explanation of the a priori status of logic
>and arithmetic. I would argue that there are transcendental preconditions on
>the very possibility of communication of structured thoughts about a shared
>reality. These are expressed in the correct logic for the operators giving
>rise to that structure. It follows as a corollary that, if we succeed in
>*evolving* a system of such structured communication, then it will embody
>as *norms* the laws in question; for they are the very precondition for the
>possibility of such a system emerging in the first place.

I hope that you now realize that the dilemma isn't mine since I was not 
trying to account for the ``a priori status of logic and 
arithmetic''---unless you mean by `` a priori'' simply their 
`unreasonable effectiveness ...''. 

I like very much your idea that there are ``transcendental preconditions 
the very possibility of communication of structured thoughts about a 
reality''. No transcendental argument that I have ever considered in 
detail was worth a hill of beans; but I hope that you will develop what 
you have in mind. My specific scepticism centers on what is built into 
the notions of `structured thought' and `shared reality'. But you must 
admit in any case that god was good to make a shared reality about which 
there could be structured thoughts. Or maybe we should say: reality is 
such that the ability to think (have structured thoughts) in a species, 
together with certain other properties, was an advantage. We can also 
hope that it continues to be.

I should mention that ``the unreasonable effectiveness of ...'' did not 
refer just to, or even primarily to, logic and arithmetic. It had to do 
with our ability to come up with kinds of mathematical structure, for 
example groups, and see them as exemplified in  natural phenomena. 

>Likewise for the concept of number. As soon as we have in place a conceptual/
>linguistic system affording reference to things, characterization of their
>properties and relations, and the resources to individuate and re-identify
>things, the concept of number is available as a *necessarily possible*
>extension of our system of referential/predicative/quantificational thought.

We are back with numbers again and, I assume, you are filling out a bit 
your transcendental deduction. But ``as soon as we have in place ...'' 
seems to me a strange way of speaking: as soon as we have the cupboards 
in place, we can put away the cups. Its as if we *did* something: 
constructed our ``conceptual/linguistic system', and then, building on 
that, our concept of number, rather than that both evolved. I do not mean 
that there was no selfconscious analysis of the structure of propositions 
that fed back into a further structuring. But we did not construct our 
language. Did our ``system of referential/predicative/quantificational 
thought really precede in evolution our conception of number---or, as I 
would prefer to say, our arithmetic?  Let me note, arithmetic was better 
developed in classical Greece than grammar and logic were.

>With apologies to Kronecker:
>	"God [i.e. transcendental constraints on the very possibility
>	of (evolved) systems of structured communication] gave us the
>	natural numbers AND logic; thereafter any rational thinker
>	really could do all the rest."

You really shouldn't apologize to Kronecker all that much. Likely your 
conception of logic would not have please him. But after all, like him, 
you left out the notions of a set of elements of a given domain and of a 
mapping between domains. And Kronecker would have been especially 
delighted that you left out of consideration, too, Cantor's theory of 
transfinite numbers. Could any rational thinker really have done all of 
that, building just on logic and arithmetic?

Let me remark briefly on Randall Holmes' posting

>But it is not appropriate to bring evolution into a discussion of the
>features of the world that reason or a hypthetical mathematical
>faculty finds in the world.  Evolution may provide an explanation of
>why we see light (it is useful to) but it provides no explanation
>whatever of why there is light or what light may be.  Similarly,
>evolution supports an explanation of why our rational or mathematical
>faculty detects and exploits intelligible structure in the world (it
>is useful to be able to do this!)

I suggested only that the theory of evolution by natural selection might 
answer the question ``why our rational or mathematical faculty is 
effective in accounting for, controlling, etc. *nature*''.

> but it does not and cannot provide
>any explanation of why there is intelligible structure in the world or
>what "intelligible structure" is.

When we speak of the ``intelligible structure of the world'' are we 
refering to anything more than whatever it is about the natural world 
that makes our science effective in dealing with it? I agree that the 
theory of evolution does not account entirely for the nature of the 

About Joe Shipman's posting, the only thing I want to remark on is:

> What mathematicians have to
>defend against social constructivists is the proposition that the integers 
>example) are not like color qualia, which neurologists successfully "reduced
>away". "Four" (or "Pi", etc.) has a stronger existence as a concept than 

Dear Joe, please don't confuse me with a social constructivist. I 
wouldn't want Jon Barwise to suspect me as a carrier of the black plague. 
A small question: it is usual to refer to `four' and `Pi' as objects 
rather than as concepts. Is there a point in your choice of words?


Bill Tait 

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