Like everyone else I read Ayer’s Language, Truth and Logic as a teenager and, like many people of a scientific bent, I loved it. The Logical Positivism that it espoused can be summarised as the claim that knowledge is of two types: (1) logical reasoning from axioms, such as used by mathematics; and (2) claims about the universe that can (in principle) be verified empirically. Anything else — such as metaphysics — is literally meaningless.That is correct up to the word "meaningless". There might be some meaning to metaphysics, but it is not the sort of knowledge that comes with a demonstration that it is true or false.
He also writes Defending scientism: mathematics is a part of science on his blog, and tries to defend these views on the Scientia philosophy blog:
I will take one statement as standing proxy for the whole of mathematics (and indeed logic). That statement is: 1 + 1 = 2I also defend logical positivism, but not this. He has abandoned the "logical" part of logical positivism. Logic and math are forms of knowledge that need no empirical verification, and usually do not get any.
Do you accept that statement as true? If so (and here I presume that you answered yes), then why?
I argue that we accept that statement as true because it works in the real world. All of our experience of the universe tells us that if you have one apple in a bag and add a second apple then you have two apples in the bag. Not three, not six and a half, not zero, but two. 1 + 1 = 2 is thus a very basic empirical fact about the world . ...
I have argued that all human knowledge is empirical and that there are no “other ways of knowing.” Further, our knowledge is a unified and seamless sphere, reflecting (as best we can discern) the unified and seamless nature of reality. ... I thus see no good reason for the claim that mathematics is a fundamentally different domain to science, with a clear epistemological demarcation between them.
Here are comments I left:
Coel, the flaw in your argument is in the triviality of your math examples. "1+1=2" is not much of a theorem, and is more accurately the definition of "2". Try applying your argument to a real theorem, such as the infinity of primes, as someone suggested.I believe that my comments and other comments refute his position.
There certainly is a sharp qualitative difference between the work of Riemann and Einstein. The mathematical theory of general relativity was worked out by Minkowski, Grossmann, Levi-Civita, and Hilbert -- all mathematicians. Einstein did not prove any theorems and rarely even made any mathematically precise statements.
Your comments about Godel's theorem are about like saying that the irrationality of the square root of 2 shattered hopes for geometry, or that comets shattered hope for astronomy. And it surely does not help your argument, unless you can explain how the theorem can be empirically understood or validated.
Your lesson from this is that "scientific results are always provisional". Maybe so, but mathematical results are not. Godel's theorem is not provisional.
You can, of course, define "science" any way you please, but you have failed to give a definition that includes mathematics. To you, science is empirical and provisional, but you do not give a single mathematical result with these properties. Do you really want to argue that "1+1=2" is a provisional result subject to empirical acceptance or rejection? Will you please tell us how this equation might be rejected?
You deny a "clear epistemological demarcation", but you do not give an example on the boundary of math and science. Your closest example is string theory, but you must realize that most of that subject is viewed by outsiders as neither science nor mathematics.
Coel, you say that it is " epistemologically identical", except that one uses empiricism and plausibility arguments and the other uses axioms and logic. In other words, not similar at all.
Part of the problem here is that what mathematicians mean by math is quite a bit different from what most scientists mean. I have heard science and engineering professors tell their students not to take math classes for math majors, because they have proofs in them. The professors act as if a proof is some sort of mysticism or voodoo with no applicability. Most of them do not understand what a proof is.
Coel argues that knowledge is science, and science is provisional, but that is just not true about mathematical knowledge. Mathematical truths are not provisional or subject to any empirical tests. He suggests that "1+1=2" can be tested by looking to see if alternate definitions can be used to predict eclipses. But there are lots of alternative number systems that are completely mathematically valid, even if they are not used to predict eclipses.
I could test "1+1=2" by laying 2 1-foot pieces of string together, and measuring the length. If so, I am likely to get 2.01 or something else not exactly 2. The mathematician says that 1+1 is exactly 2. So what have you tested? You certainly have not validated the mathematical truth that 1+1 is exactly 2. You have an empirical result about the usefulness of the equation, and that's all.
So when Coel says that math is epistemologically identical to science, he is not talking about how mathematicians do math.
SciSal is correct that most math has nothing to do with modeling the real world.
String theory is an odd beast. There are some mathematicians who prove theorems about string models, and physicists who look for empirical tests. But the vast majority of string theorists are not concerned with either of these pursuits. They are more like people playing Dungeons & Dragons in their own imaginary universe.
Coel, you say that math axioms are codified regularities of nature. The most common axiom system for math is ZFC (Zermelo-Frankel). Can you explain how those axioms relate to nature?
Coel, you repeatedly deny any distinction between a definition, a theorem, and an equation that empirically seems approximately valid. So I would lump you in with those other science and engineering professors who do not recognize the value of a proof.