> The universe was here before people existed. But it didn't operate
> according to mathematical laws, it just operated, period. Laws (unlike
> Plato's Forms), however, exist only in human brains; they represent our
> collective effort (never complete, never exact) to understand that
> operating of the universe. No human brains, no laws, mathematical or
> otherwise. It has been a long time since I read the _Anti-Duhring_, but
> as I understood it at the time, Engels held to this position: i.e., he
> held that the belief in the non-mental existence of laws led to
> idealism. The universe is extra-mental; laws are mental.
OK, let's try it this way. For a long time before human beings existed, the moon orbited the earth, and the earth and the other planets orbited the sun in basically the same way they do now -- right? Since Newton, we have been able to describe this behavior mathematically using his law of gravitation: the force by which two masses attract each other is proportional to the product of their masses divided by the square of the distance between them -- right? So even before human beings existed, *these masses behaved according to Newton's Law.* (I would put these words in caps and double-underscore them, but I want to send a plain-text e-mail, and all-caps isn't polite.) That law was *not* invented by Newton or any other human beings -- right?
I think the following quote puts my view better than I could:
"The realisation that most scientific observations are context-dependent has led some philosophers to argue that science is a social construct which has nothing to do with reality and is solely a matter of human convention. This argument stems from the entirely sensible modern perception that scientific 'truth' is not absolute, but depends upon having some agreed common conceptual framework. However, the belief that science is solely a construct, which by implication could be whatever scientists decided to agree on, is really very silly - however elegantly phrased - because it ignores a very important aspect of these conceptual frameworks. They are not arbitary: they are the outcome of a previous scientific process. For example, scientists cannot make objects float skywards merely by agreeing amongst themselves that the force of gravity acts up rather than down." (Stewart, I., & Cohen, J. (1997) "Figments of reality: the evolution of the curious mind", Cambridge University Press, Cambridge) [quoted at http://homepages.which.net/~gk.sherman/gcaa.htm]
In a very small nutshell: to gain knowledge, we ask questions and get answers. Science asks nature questions; nature gives the answers. We can ask any questions we want -- depending on the era, the society, even the individual -- but we do *not* get to make up the answers. Nature gives us the answers, whether we like them or not.
This nutshell needs to be "unpacked," as philosophers like to say, and developed at quite some length. That has been done in numerous books and papers. For example, one very good anthology on the subject I happen to have on hand is Martin Curd & J. A. Cover: _Philosophy of Science: The Central Issues_, New York/London, W. W. Norton & Co., 1998. It is chock-full of informative stuff about the philosophy of science, including very penetrating discussions of this "scientists make everything up based on their culture" nonsense. But it's over 1000 pages, and pretty hard going for someone without philosophical training.
A lot shorter and easier-to-follow work I also have with me is John Losee, _A Historical Introduction to the Philosophy of Science_, Oxford/New York, Oxford UP, 1993. A very useful general reference covering philosophy of science, philosophy of mathematics, philosophy of logic, etc., etc., is _The Blackwell Companion to Phiosophy_, ed. Nicholas Bunnin and E. P. Tsui-James, Oxford/Cambridge, Blackwell Publishers, 1996.)
> P.S. I don't know if this is relevant or not. The ratio of the
> circumference to the radius of a circle is _not_ equal to any value of
> pi that has ever been expressed. And in fact the mathematician's circle
> does not exist in extra-mental reality; it is only an ideal abstraction
> from the 'circles' we see in the world. Mathematics only approximates
> the real world; but the real world is not an approximation to itself.
> It
> merely is.
Not quite. The value of pi *is* the ratio of the circumference of a circle to its diameter (*not* its radius). This number is a transcendental number, i.e., it is not the root of any polynomial equation (see http://en.wikipedia.org/wiki/Transcendental_number). This fact (I emphasize *fact*) was proved in 1882 by Ferdinand von Lindemann; the proof is called the Lindemann-Weierstrass theorem (see http://en.wikipedia.org/wiki/Lindemann-Weierstrass_theorem). So pi is in fact equal to circumference/diameter; it's just that, as a transcendental and therefore irrational number, it can't be expressed as a decimal with a finite number of decimal places. That's where the approximation comes in; it has to do with the difference between rational and irrational numbers. It doesn't have anything to do with the relationship between mathematics and the real world, whatever that is; it's entirely within mathematics.
Now, the question you raise is whether the "mathematician's circle" exists in an "extra-mental reality." This is a very big topic in the philosophy of mathematics (see http://en.wikipedia.org/wiki/Philosophy_of_mathematics) to this day, and still controversial. What can perhaps be said fairly safely to begin with is that mathematics is a science which can be (at least partially) formalized (see http://en.wikipedia.org/wiki/Formal_system). That is, one gives definitions of the terms used in it, and also axioms, which are not proven, and proves the rest of the system by deriving theorems from the axioms by the usual logical rules of deduction.
However, as Godel showed (see http://en.wikipedia.org/wiki/Godel%27s_proof), not every true statement even in rather elementary parts of mathematics can be reached by such a formalization process. So what is the rest of it? If this question interests you, you could start by consulting the Wikipedia page on the philosophy of mathematics, but the chapter on philosophy of mathematics in the Blackwell Companion I referred to above is much better.
In short, I think people who want to get involved in this discussion need to do a little homework first.
Jon Johanning // jjohanning at igc.org __________________________
From Translation all science had its offspring.
-Giordano Bruno (quoted by John Florio, 1603)