Best book for understanding arithmetic?

[fresh_42]

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

@Antisthenes
In case you've read Kant or Habermas in the original version, it might be worth to switch the language of those Wikipedia pages, although I haven't checked to which extend they differ. Usually they do quite a lot.

 

[Antisthenes]

@Buffu

Yes, can do that of course. But thought perhaps that arithmetic is much more than that, and that a math imbecile like me needs as much knowledge in elementary math as possible, before trying to master the art and creativity of math on higher levels.

@fresh_42

Good advice, tnx :) However, is there perhaps more books, for newbies, about the general mindset necessary for being a good mathematician? Read "A Mind for Numbers" by Oakley, but it contained almost no examples of how to solve math problems in creative and different ways. Really need to learn flexibility when visualising and thinking mathematically, it seems.

 

[Buffu]

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.

Yes, can do that of course. But thought perhaps that arithmetic is much more than that, and that a math imbecile like me needs as much knowledge in elementary math as possible

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

before trying to master the art and creativity of math on higher levels.

Arithmetic (+,-,/,*) is of very little use in maths. If you want to learn maths then you should learn algebra/calculus and leave all your arithmetic (+,-,/,*) to Stephen Wolfram.

 

[fresh_42]

However, is there perhaps more books, for newbies, about the general mindset necessary for being a good mathematician? Read "A Mind for Numbers" by Oakley, but it contained almost no examples of how to solve math problems in creative and different ways. Really need to learn flexibility when visualising and thinking mathematically, it seems.

Oversimplified we may say, that mathematics is a bit like playing LEGO, including the player's character. The textbooks can be compared with the accurate and detailed models of all kinds, from castles to space ships. Doing mathematics is then the big box which contains all kinds of bricks and you start building something without a detailed plan but a general idea of the result. Until then the structure is rebuilt many times, extended, changed and often a compromise due to the lack of optimal bricks. So the bigger your box is, i.e. the more deconstructed models are in it - the more textbooks you've read, the better you can build whatever you want to.

I would say: Grab an algebra book (van der Waerden as I quoted in post #11, if you like more text, or Serge Lang if you prefer more formulas, or a freely available version like in openstacks, cp. posts #2, #9 and the link there), then a book on calculus (usually people here recommend Spivak, but I don't know, since my elementary ones aren't in English, or which I like very much, Hewitt, Stromberg, (in engl.) especially because it contains a large section about the axiom of choice at the beginning, but gets very elaborated quickly, or again openstacks for free) and either read both of them or evaluate which kind of mathematics comes easier to you, because these are the main streams with different techniques.

 

[fresh_42]

I won't call that arithmetic.

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.

 

[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 Schrödinger'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