
The Principles of Mathematics (1903) - osivertsson
http://people.umass.edu/klement/pom/pom.html
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orangutango
"[A]ll pure mathematics deals exclusively with concepts definable in terms of
a very small number of fundamental logical concepts, and that all its
propositions are deducible from a very small number of fundamental logical
principles ..."

Cue Gödel... [1]

[1]
[https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompletenes...](https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)

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hackinthebochs
All pure math can be deducible from axioms doesn't mean that all math can be
deducible from a single set of axioms. Rather, it means that for each
mathematical proposition there is a set of axioms from which one can deduce
it.

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orangutango
For any set of axioms you take, if they are consistent then it is incomplete.
You can enhance the axiom set to extend its reach, but an adversary can always
find true, unprovable statements.

My main point is I disagree with the view of mathematics as nothing more than
some axiomatic program-- in 1903 many were hopeful that a system (like
Russell's formal logic in Principia) could simply generate the truths of
mathematics. Gödel shattered that dream.

~~~
hackinthebochs
>My main point is I disagree with the view of mathematics as nothing more than
some axiomatic program

I just don't see how this follows from Godel. It gives us a more expansive
view of math, but I don't see how any fundamental understanding is overturned.
I don't see how this takes away from the connection between axioms and
theorems. The characterization of math as discovering the logical consequences
of axioms is just as true.

~~~
johnbender
Cousin comment helped me out a bunch:

[https://news.ycombinator.com/item?id=14540054](https://news.ycombinator.com/item?id=14540054)

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aaachilless
Is this level of density typical for some subset of advanced maths texts? I've
never seen anything like it and I'm about to receive an undergrad maths
degree.

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j2kun
This appears to be a philosophy text, to my cursory glance. Definitely not
like the advanced math texts I have read.

~~~
aphextron
>This appears to be a philosophy text, to my cursory glance. Definitely not
like the advanced math texts I have read.

It is absolutely not a philosophy text. It's an attempt at defining a rigorous
definition of the basic axioms we take for granted in mathematics in terms of
pure logic, and then examining whether those basic rules of logic are
themselves irreducible forms of nature or further creations of man.
Essentially it seeks to support all higher mathematical thought by not just
taking axioms for granted, but formally proving every one.

~~~
j2kun
You say that, and I'm aware of the historical context. But you cannot deny
that the majority of the text is devoted to questions like which verbs are
appropriate and how words as used in mathematics relate to their notions used
in math.

In my experience, logic falls into two catgeories: using math to study a
formal system of logic; and determining "foundations for all math" and
determining what is an "irreducible form of nature" and other such prosaic
things (can the phrase "such that" be defined?). In my humble opinion, the
former is mathematics and the latter philosophy.

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fibo
I have this book, I recommend it especially if you like math and you are a
teenager.

