Comments on “Positive and logical”

Logical positivism (resp. logical empiricism) is concerned with statements that van be verified logically (resp. empirically). It does not say all statements can be verified. So Goedel’s Incompleteness results just show there are certain statements which cannot be verified, i.e. which are not subject to logical positivism and/or empiricism. Also, Incompleteness is not a flaw in classical logical logic, rather it is just an aspect of that logical system. So I don’t see how you can claim, classical logic is “inherently somewhat broken”.

If something is mathematically true, you can be absolutely sure of it, because a mathematical proof is unarguably correct.

This is true.

There are mathematical truths that can’t be proven.

This is false. If a mathematical statement cannot be proven, then it it is not a mathematical truth. Mathematical truths are, tautologically, mathematical statements for which a proof has been given.

The promise of rationalism was that, by rational methods, we can eventually come to know anything that is true.

Rationalism doesn’t assert that all claims are subject to verification by logical reasoning, but that only statements that are subject to verification by logical reasoning can be said to be true.

proving mathematics correct—also failed.

I’m not sure what this is trying to say. Mathematics certainly has issues, namely the prevalence of classical logic as opposed to intuitionism, which limits its practical applicability. As do the usage of set theoretic foundations over type theoretic ones. But if you fix your base logic and axioms then everything derived therefrom is correct. You obviously can’t prove the axioms or they would be theorems rather than axioms.

The idea that logical positivism “failed” is only possible in fields like philosophy where there is no expectation of logical rigor. It was merely generalized. Science today is concerned with statements subject to refutability. Logical positivism/empiricism is concerned with statements subject to verifiability, which are a fortiori subject to refutability.

Of course rationalism “failed” if you define it to mean a system that claims everything is subject to understanding via reason. But that’s disingenuous. Rationalism is just the view that knowledge of truth derives from reason. Philosophers seem not to like this because it points out that topics like metaphysics are inherently worthless. Obviously knowledge can be derived from empiricism as well, but such knowledge need only approximate truth.

Hello Dr. Chapman,

Thanks for writing this fantastic book. I really needed this as I was going through a phase of post-rationalist nihilism after leaving LessWrong. I was wondering if you have a meta-rationality reading list for exploring these topics further?

Regards,
Muhammad

I would like to mention Imre Lakatos’ Proofs and Refutations. Here is a link to the corresponding Wikipedia article.
It is a wonderful essay describing the nebulosity of mathematics.