Best book for understanding arithmetic?

[Buffu]

Hilbert (and the mathematicians and logicians of his time) called it arithmetic. Who am I to correct those Grands? But it may well be a matter of language here. E.g. we do not use the word algebra for simple calculations. Or calculus for analysis. Arithmetic involves everything which mathematics is build upon, so yes it involves logic and set theory.

Hilbert's language looks obsolete to me.

Does OP wants to learn logic or arithmetic(+,-,/,*) ?

 

[fresh_42]

Hilbert's language looks obsolete to me.
Does OP wants to learn logic or arithmetic(+,-,/,*) ?

And the English terms look inexact and vague to me, so we both have our burden. What you call arithmetic doesn't even qualify as mathematics in my view. It is calculations, not math.

You can't have arithmetic without logic, at least not, if you want to understand it, or you are at least a bit interested in history and what is known as the big crisis of mathematics. Physicists are still struggling with theirs, which is the impression I have, when I read in the QM forum.

 

[gmax137]

This is an odd but interesting thread.

Two years ago my ambition was to learn quantum physics, and could easily follow math explanations all the way up to the first university level, but hit a wall when trying to solve problems myself. So now it's back to basics. Just want to understand the foundation of arithmetic properly, and then take it from there, if I have the IQ.

especially because it contains a large section about the axiom of choice at the beginning

Quantum mechanics problem sets are difficult for everybody -- by design. The only way to get better at solving the problems is to do more problems. I highly doubt that deeper study of the axiom of choice is going to help someone struggling with Schrodinger's equation for a potential well.

 

[Mark44]

Can you add, subtract, multiply and divide ? There is nothing more to arithmetic than that.

Not from the logical point of view:
https://en.wikipedia.org/wiki/Axiomatic_system
https://en.wikipedia.org/wiki/Deduction_theorem
https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

I won't call that arithmetic. That looks like Mathematical logic.

I agree with @Buffu here. In common usage, arithmetic is just addition, subtraction, multiplication, and division.

If you already know arithmetic(+,-,/,*) then what else are going to learn in it, you already know everything.

Everything about arithmetic ...
I agree.

 

[Mark44]

You can't have arithmetic without logic, at least not, if you want to understand it,

I don't believe logic is a prerequisite for learning arithmetic. When I was learning how to add/subtract/multiply/divide numbers (including fractions and decimals) there was no mention of logic whatsoever. To understand why $\frac 1 2 + \frac 1 2 = 1$ requires nothing more than the field axioms (and specifically the distributive law), which I mentioned earlier (and which were presented much later). Slightly more complicated additions such as $\frac 1 3 + \frac 1 2$ require slightly more work, namely multiplying each fraction by 1 in a suitable form (multiplication by 1 is an axiom that I also mentioned before) followed by use of the distributive law again. If you're arguing that the field axioms are logic, that's something of a stretch. Certainly one can understand arithmetic without knowing anything about Godel or incompleteness. When you were learning arithmetic, was there any mention of an Incompleteness Theorem? I thought not.

The OP indicates that he wants to learn arithmetic. If he can add, subtract, multiply, and divide real numbers, then all is good, and he should move on to something more advanced that arithmetic.

 

[fresh_42]

In common usage, arithmetic is just addition, subtraction, multiplication, and division.

I don't call this mathematics. This is counting.

But I do admit that I'm an old fashioned traditionalist: (https://en.wikipedia.org/wiki/Arithmetic)

The terms arithmetic and higher arithmetic were used until the beginning of the 20th century as synonyms for number theory and are sometimes still used to refer to a wider part of number theory.

And as I said, it might be a matter of language.

 

[Buffu]

What you call arithmetic doesn't even qualify as mathematics in my view. It is calculations, not math.

That's what I mean when I said arithmetic(+,-,*,/) is for Stephen Wolfram.

You can't have arithmetic without logic, at least not, if you want to understand it, or you are at least a bit interested in history and what is known as the big crisis of mathematics. Physicists are still struggling with theirs, which is the impression I have, when I read in the QM forum.

But elementary school students don't learn from 500 page undergrad logic book.I am still confused, what does OP wants to learn ? Logic or arithmetic (+,*,/,-) which I guess he already knows.

 

[Mark44]

I don't call this mathematics. This is counting.

See below.

But I do admit that I'm an old fashioned traditionalist: (https://en.wikipedia.org/wiki/Arithmetic)

And as I said, it might be a matter of language.

From the wiki article you linked to, first paragraph (emphasis added by me):

Arithmetic [...] is a branch of mathematics that consists of the study of numbers, especially the properties of the traditional operations on them: addition, subtraction, multiplication, and division.

Later in the same paragraph it mentions that up until the early 1900s, arithmetic and higher arithmetic were synonyms for number theory, but that's close to 120 years ago.

 

[fresh_42]

I am still confused, what does OP wants to learn ? Logic or arithmetic (+,*,/,-) which I guess he already knows.

Yes, I agree, it seems a bit hard to figure out. That's why I suggested (in post #33) to read an algebra and a calculus book and see what fits. On the other hand, he said he's familiar with philosophy, which automatically (in my opinion) brings Russell, Cantor, Zermelo, Gödel and logic at the table.

 

[fresh_42]

I think, we can agree to disagree here.

 

[Mark44]

Yes, I agree, it seems a bit hard to figure out. That's why I suggested (in post #33) to read an algebra and a calculus book and see what fits.

That's my advice (and @Buffu's) as well.

On the other hand, he said he's familiar with philosophy, which automatically (in my opinion) brings Russell, Cantor, Zermelo, Gödel and logic at the table.

Hard to see that this will help him with arithmetic, which is the goal that he has repeatedly stated.

 

[Antisthenes]

The disagreement helps me to get a better perspective of arithmetic, so appreciate the input from all of you. Being a philosopher I need math because I want to better understand the origins of three things: the universe, life and consciousness. Of course, this is a large canvas that no single human being can cover in a lifetime, especially not with my lack of math talent. But it makes it clear that spending unnecessary time on arithmetic is not wise in this case. Will therefore start to study elementary algebra, and perhaps eventually understand higher levels too, now that I have accepted that math is not only logic but also creative art that demands a lot of practice, like all art forms.

 

[Mark44]

But it makes it clear that spending unnecessary time on arithmetic is not wise in this case. Will therefore start to study elementary algebra, and perhaps eventually understand higher levels too, now that I have accepted that math is not only logic but also creative art that demands a lot of practice, like all art forms.

Excellent choice. Good luck in your studies!

 

[Antisthenes]

Thanks :) Will still try to have the same attitude toward algebra: practice and enjoy it for it's own sake. One day at a time. Then one night I might discover in a distant future that I gradually and indirectly built the foundation to understand linear algebra in quantum physics, for example.

 

[Buffu]

Russell, Cantor, Zermelo, Gödel

Russell, Cantor, Zermelo, Gödel += Frege, Whitehead.

 

[mathwonk]

for another perspective on what arithmetic is, see this book by J.P. Serre, Fields medalist: it includes quadratic forms, modular forms, Gaussian reciprocity, group characters and representations, and Dirchlet's theorem on primes in arithmetic progressions via complex analysis and analytic continuation. the goal of the last result being to answer for instance the question: since primes greater than 5 must end in either 1,3,7, or 9, is any of these endings preferred, or do they all occur essentially the same number of times?

https://www.amazon.com/Course-Arithmetic-Graduate-Texts-Mathematics/dp/0387900403/ref=sr_1_1?s=books&ie=UTF8&qid=1515714781&sr=1-1&keywords=serre,+course+in+arithmetic