The Principles of Mathematics (1903)
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The Principles of Mathematics (1903)
1–10 of 22 posts
Re: The Principles of Mathematics (1903)
#2"[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...
Re: The Principles of Mathematics (1903)
#3Is 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.
Re: The Principles of Mathematics (1903)
#4"[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...
Incompleteness means there are true statements that can't be proven. Given that any standard set of "fundamental logical concepts" is probably sound and as long as "all pure mathematics" means "that which can be proven" then there's nothing wrong with saying that "all its propositions are deducible" from those principles.
Re: The Principles of Mathematics (1903)
#5Is 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.
This appears to be a philosophy text, to my cursory glance. Definitely not like the advanced math texts I have read.
Re: The Principles of Mathematics (1903)
#6Is 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.
this was written in 1903 by bertrand russell who was both a philosopher and a writer which garnered a nobel prize in literature.. russell loved words
i am almost positive all of the views recited in this piece where still considered relevant today have since been made more accessible from textbooks to youtube videos
do you need to, or even need to want to, read this piece by russell? hardly, but at the same time it was written by a well respected individual in the field and there could be value in trying to dissect the text
regardless of your views on any single piece of writing i would wholeheartedly encourage your interest in mathematics
much like how i would encourage someone who dislikes or finds shakespeare too dense to still pursue a career in writing if they so desired but i would still recommend them to take another look
Re: The Principles of Mathematics (1903)
#7"[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...
Incompleteness means there are true statements that can't be proven. Given that any standard set of "fundamental logical concepts" is probably sound and as long as "all pure mathematics" means "that which can be proven" then there's nothing wrong with saying that "all its propositions are deducible" from those principles.
I don't think this is a correct view of what the Incompleteness Theorem says. The Incompleteness Theorem says that there are statements that are true in the standard model of the integers that are not provable in the first order Peano axioms. This does not mean that such statements are not provable. They just aren't provable in the first order axioms. The second order axioms are categorical and this means using the second order axioms any true statement can be proven.
The second order Peano axioms are a superset of the first order axioms. There is one small change in one of the axioms and that is the difference between the two systems.
Re: The Principles of Mathematics (1903)
#8"[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...
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.
Re: The Principles of Mathematics (1903)
#9Is 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.
This appears to be a philosophy text, to my cursory glance. Definitely not like the advanced math texts I have read.
>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.
Re: The Principles of Mathematics (1903)
#10Earlier quoted context omitted.
This appears to be a philosophy text, to my cursory glance. Definitely not like the advanced math texts I have read.
>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. Essenti…
This is not a nitpick, I'm actually curious: When you say "absolutely not a philosophy text" do you mean that it's definitely not 100% philosophy? Or do you mean that it's actually 0% philosophy?