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**General Science / Re: Does All Of Mathematics Come Down To Simple Arithmetic?**

« **on:**

**Yesterday**at 22:52:38 »

Hi.

(I started this several times and deleted it. I finally decided I might as well post something, convsersation has been very light at home lately. It was supposed to be short but then it got late and I lost my way, sorry).

One odd thing is that @Jimbee seems to make an opening post and then never makes another post on the same thread. (I've checked the most recent 10 posts listed on your profile, @Jimbee ). You are most welcome to join in the discussion at any time, it's a forum. Otherwise, a few of us will just use the space to discuss whatever topics arise no matter how tangential to the OP they have become. That's also OK but if you want something specific discussed or more info on anything that has been raised then you can say so.

I'm not going to start anything new. I will just comment on some bits that have already appeared.

A lot of people who have only studied the compulsory bit of Maths at school may think that Maths is just arithmetic and so more advanced Mathematics must just be arithmetic but the sums are longer. Somewhere at the top end of school or early University you may learn and accept that "the Mathematics" wasn't about how numbers combine. They were just an example of something with a suitable algebraic structure. A bit further along in University you may study set theory and learn or accept that everything you've been doing in Mathematics is just set theory and all mathematical objects are just sets. ( @Bored chemist was the first one to imply set theory is important by mentioning the 1+1=2 proof and then @evan_au mentioned

Near the top end of University you may learn something else. The modern thinking is that sets and set theory aren't the only place you could start from. For example, we can obtain all the results of an advanced topic in Mathematics like "representation theory" starting from "set theory". However we can also obtain set theory from representation theory. Therefore, it is quite arbitrary whether you consider sets (and set theory) or representations (and representation theory) as your most fundamental or atomic objects from which all the rest of Mathematics can be built.

Indeed, there may be many places that you could start from and consider as your most fundamental objects. Some popular choices inlcude starting from geometry, or from the notion that mathematics can be built up from a set of fudamental or atomic algorithms.

Anyway, why does this matter? Well, it means that you probably could start from the Natural numbers and arithmetic. On the face of it that seems to offer some good algebraic structure to begin with and you probably CAN build all the rest of Mathematics from this. We seem to have turned a full circle, the notion people may have had at school - that all Mathematics is just arithmetic - may actually be right.

It's probably time to address some of the other posts that have been made:

You may think: f is a function. This means it's a rule taking a number to another number. For example, the rule might be to double the number and we can write f(1) =2, f(2) = 4, ... f(6)=12,.... and in general f(x) = 2x.

It's important to note that f isn't the image you obtain, f is "the rule" that maps a number to its image, we have some notation that really attempts to make this clear and instead of defining the function as f(x)=2x we can write

f : x --> 2x

So how is this mathematical object, a function, actually a set?

f is this set,

f = { (1,2), (2,4), ...... , (6,12), ...... }

in words it is the set of ordered pairs of numbers like (1,2) and (2,4) where the first ordinate is a number in the domain of f and the second ordinate is the corresponding image under f.

Where we write f(1)=2 that has just been short-hand for writing (1,2) ∈ f

(in words, (1,2) is a member of f ).

Note that an ordered pair like (a,b) can be written as a simple set.

(for brevity, I'll just provide a reference: https://en.wikipedia.org/wiki/Ordered_pair#Kuratowski's_definition - there are actually many ways to define an ordered pair but Kuratowski's definition is widely used).

Anyway, the point is that f is a set and it's elements are just sets. It's all sets. Notation like f(x) = 2x has always just been shorthand for the set theoretic expression ∀x( (x,2x) ∈ f) although it's easier to read if we just write out the set f this way,

f = { (x,2x) : x ∈

The keen eyed may notice that there is still something in these sets, e.g. the ordered pair (a,b) has the element {

I've taken the time to set out clearly that a function is a set because it helps to make the jump to seeing that every, absolutely every, mathematical object is a set. Numbers, functions, vector spaces, modules, whatever mathematical object you can think of.... they're ALL just sets.

(..... well, anyway.... that's some hint of how it feels when you finally study set theory at University... in modern parlance it's an " o.m.ds. - we've actually found our atoms" moment).

I'm not going try and get from Natural numbers and arithmetic back to sets and set theory, this has probably been long and boring enough. I'll just repeat what was said above - you probably could start from Natural numbers and arithmetic and take those as your atoms.

Of course what @paul cotter actually said was the following....

By the way, considering Mathematics as a verb is very good way to consider what Mathematics is. Unlike a real scientist, mathematicians do very little. There is nothing new they can find, like a new trans-uranic element, all of Mathematics is already there and could have been seen. A mathematician is just someone who can demonstrate that one thing is the same as another thing in very small steps that another human being can follow. A smarter being would have just seen it and made the jump in one go. The common phrase is that God would be a poor Mathematician because he has no need for it.

Taylor series or Power Series. The square root function is just a sum of terms, each term involves only integer powers - which are clearly just based on (an integer number of) multiplications .... multiplication being based on addition. It's all just addition and subtraction.

(i) Subtraction is not commutative. 5 -1 ≠ 1 - 5.

(ii) The integers do not form a field. A requirement for a field is that multiplicative inverses exist for every non-zero element. An integer like 5 has the fraction 1/5 as a multiplicative inverse but sadly this fraction isn't in the integers. The Rationals would form a field, the integers won't. This is important because the integers should be recognised as being "incomplete". It's an example of how human beings did manage to develop more advanced and idealised algebraic structures even if they started from a place that was littered with imperfect and incomplete structure.

(iii) You imply cause and effect where none exists. "A field" is just the name we give to this sort of structure not any explanation for it. "Light travels at the speed of light because it's a luxon" offers no explanation for why light travels at this speed, the term "luxon" is just the name we give to a particle travelling at that speed.

Seriously, @evan_au, you write very well and spend the time to put in references and hyperlinks. Most of the time they're the best and most easily read and understood things in every thread. People may read your posts, so in the future you just can't post if it gets too late.

As breifly hinted at above, Mathematics may be very circular. There may be many places you could start from to build up all the rest of Mathematics.

Is it meaningless?

Now, I wouldn't want anyone to think that Mathematics must be junk because anything based on a circular argument is usually junk. Quite the contrary, it may be one of the strengths of Mathematics and provide some assurance that it is something worth studying.

No matter what you start from and what you consider as your fundamental objects, you'll probably end up being able to re-create the entirity of Mathematics. So, for example, aliens on planet A who are more interested in some geometry that appears in their environment instead of sets should still end up with the same complete works of Mathematics that we have. Other aliens on planet B may be completely blind, they can't see any shapes, their main sense might be feeling vibrations (like a spider on a web). Studying the harmonics that can exist on a string will probably also allow them to develop the entirity of Mathematics, even though the natural number system they learn in school may seem very alien to us. For example a string vibrating with the first harmonic could be driven again at the same frequency and all that happens is that the amplitude gets bigger, the frequency doesn't change. So, for them, their logical number-like system is 1 + 1 = 1 If we said to them 1 + 1 = 2 they might think we were idiots who never went to school. However, their Mathematicians will realise that we are just describing a slightly different structure: If you join two strings together and each was vibrating with it's 1st harmonic, then you can end up with a new string (of twice the length) and that would be vibrating with its 2nd harmonic frequency, so they will understand that we are just using a different algebraic structure. Anyway, hopefully you get the point... all algebraic structures and all of mathematics could be re-created by studying the vibrations on a string.

Of course, this tacitly assumes that several things in nature do follow or have some structure which we may call an algebraic structure. Well.... it seems that it does but that's a discussion the scientists can have in another thread if they wish. Historically, mathematics was rooted in some stuff that seemed to appear in nature. Later on it's possible to just propose some new axiom system that may have no basis in nature or reason to assume its truth and still declare that you are perfoming or doing some Mathematics. However, I would still argue that this is justified by some structure existing or being exhibited in Nature. Specifically, we can study an arbitrary structure and axiom system because we can argue that there is some Mathematical system that does seem to hold in Nature, so it's sensible to conisder "what if" scenarios..... what if there were other axioms and other structures?

It's too late and too long already. I'm signing off.

Best Wishes.

(I started this several times and deleted it. I finally decided I might as well post something, convsersation has been very light at home lately. It was supposed to be short but then it got late and I lost my way, sorry).

One odd thing is that @Jimbee seems to make an opening post and then never makes another post on the same thread. (I've checked the most recent 10 posts listed on your profile, @Jimbee ). You are most welcome to join in the discussion at any time, it's a forum. Otherwise, a few of us will just use the space to discuss whatever topics arise no matter how tangential to the OP they have become. That's also OK but if you want something specific discussed or more info on anything that has been raised then you can say so.

I'm not going to start anything new. I will just comment on some bits that have already appeared.

Someone on another message board once said, that all of science is reducible down to math.In the UK we condense the word Mathematics to

**Maths**instead of Math. It may look silly when you analyse it, since it would seem that Maths is a plural form and implies there are many different Maths you could use. Well there probably is and at the same time it's all something with a lot in common, as implied by others ( @varsigma 's post is most closely related).A lot of people who have only studied the compulsory bit of Maths at school may think that Maths is just arithmetic and so more advanced Mathematics must just be arithmetic but the sums are longer. Somewhere at the top end of school or early University you may learn and accept that "the Mathematics" wasn't about how numbers combine. They were just an example of something with a suitable algebraic structure. A bit further along in University you may study set theory and learn or accept that everything you've been doing in Mathematics is just set theory and all mathematical objects are just sets. ( @Bored chemist was the first one to imply set theory is important by mentioning the 1+1=2 proof and then @evan_au mentioned

**Peano's axioms**directly in his post soon afterwards, which you can Google if you wanted more information). You will probably be both surprised (or disappointed?) to realise that everything you've done was based on something so simple but also quite pleased that at least now you know what it is you've been doing and working with. You'll be fairly happy knowing that in Mathematics "sets" are the equivalent of "atoms" in Physics, they are the most fundamental object and everything else is just built up from them.Near the top end of University you may learn something else. The modern thinking is that sets and set theory aren't the only place you could start from. For example, we can obtain all the results of an advanced topic in Mathematics like "representation theory" starting from "set theory". However we can also obtain set theory from representation theory. Therefore, it is quite arbitrary whether you consider sets (and set theory) or representations (and representation theory) as your most fundamental or atomic objects from which all the rest of Mathematics can be built.

Indeed, there may be many places that you could start from and consider as your most fundamental objects. Some popular choices inlcude starting from geometry, or from the notion that mathematics can be built up from a set of fudamental or atomic algorithms.

Anyway, why does this matter? Well, it means that you probably could start from the Natural numbers and arithmetic. On the face of it that seems to offer some good algebraic structure to begin with and you probably CAN build all the rest of Mathematics from this. We seem to have turned a full circle, the notion people may have had at school - that all Mathematics is just arithmetic - may actually be right.

It's probably time to address some of the other posts that have been made:

Not really, once you get into calculus you need to understand the concepts of limits and functions.It would take a lot of lines to define or construct limits in terms of sets but functions we can do very quickly.

You may think: f is a function. This means it's a rule taking a number to another number. For example, the rule might be to double the number and we can write f(1) =2, f(2) = 4, ... f(6)=12,.... and in general f(x) = 2x.

It's important to note that f isn't the image you obtain, f is "the rule" that maps a number to its image, we have some notation that really attempts to make this clear and instead of defining the function as f(x)=2x we can write

f : x --> 2x

So how is this mathematical object, a function, actually a set?

f is this set,

f = { (1,2), (2,4), ...... , (6,12), ...... }

in words it is the set of ordered pairs of numbers like (1,2) and (2,4) where the first ordinate is a number in the domain of f and the second ordinate is the corresponding image under f.

Where we write f(1)=2 that has just been short-hand for writing (1,2) ∈ f

(in words, (1,2) is a member of f ).

Note that an ordered pair like (a,b) can be written as a simple set.

(for brevity, I'll just provide a reference: https://en.wikipedia.org/wiki/Ordered_pair#Kuratowski's_definition - there are actually many ways to define an ordered pair but Kuratowski's definition is widely used).

Anyway, the point is that f is a set and it's elements are just sets. It's all sets. Notation like f(x) = 2x has always just been shorthand for the set theoretic expression ∀x( (x,2x) ∈ f) although it's easier to read if we just write out the set f this way,

f = { (x,2x) : x ∈

**Z**}The keen eyed may notice that there is still something in these sets, e.g. the ordered pair (a,b) has the element {

*} in it. This means that***a***must be something, perhaps we still need the natural numbers to start from. One of the things that Peano's axioms shows us is that we can construct the Natural numbers just from the empty set, ∅. So, it turns that everything can be a set and we need only the most basic of sets, the empty set, as the thing we start from.***a**I've taken the time to set out clearly that a function is a set because it helps to make the jump to seeing that every, absolutely every, mathematical object is a set. Numbers, functions, vector spaces, modules, whatever mathematical object you can think of.... they're ALL just sets.

(..... well, anyway.... that's some hint of how it feels when you finally study set theory at University... in modern parlance it's an " o.m.ds. - we've actually found our atoms" moment).

I'm not going try and get from Natural numbers and arithmetic back to sets and set theory, this has probably been long and boring enough. I'll just repeat what was said above - you probably could start from Natural numbers and arithmetic and take those as your atoms.

Of course what @paul cotter actually said was the following....

.... you need to understand the concepts....This is regarding Mathematics as something a human being does, treating "Mathematics" as a verb rather than as a noun. If we treat it as a noun, then Limits and functions will exist irrespective of whether some human being understands the concepts. There is a human need to understand some concepts that don't seem to be like arithmetic because we are not computers and any attempt to set this out as arithmetic could take years.

By the way, considering Mathematics as a verb is very good way to consider what Mathematics is. Unlike a real scientist, mathematicians do very little. There is nothing new they can find, like a new trans-uranic element, all of Mathematics is already there and could have been seen. A mathematician is just someone who can demonstrate that one thing is the same as another thing in very small steps that another human being can follow. A smarter being would have just seen it and made the jump in one go. The common phrase is that God would be a poor Mathematician because he has no need for it.

... and the square root function doesn't break down into addition and subtraction....

Taylor series or Power Series. The square root function is just a sum of terms, each term involves only integer powers - which are clearly just based on (an integer number of) multiplications .... multiplication being based on addition. It's all just addition and subtraction.

Similarly, you could write code for square roots,.....Indeed, for any analytical function because even if nothing easier presents itself, you'd still always have power series.

The familiar "Commutative" property of addition, subtraction and multiplication is a result of the integers forming a mathematical "Field".Was it late when you wrote that?

(i) Subtraction is not commutative. 5 -1 ≠ 1 - 5.

(ii) The integers do not form a field. A requirement for a field is that multiplicative inverses exist for every non-zero element. An integer like 5 has the fraction 1/5 as a multiplicative inverse but sadly this fraction isn't in the integers. The Rationals would form a field, the integers won't. This is important because the integers should be recognised as being "incomplete". It's an example of how human beings did manage to develop more advanced and idealised algebraic structures even if they started from a place that was littered with imperfect and incomplete structure.

(iii) You imply cause and effect where none exists. "A field" is just the name we give to this sort of structure not any explanation for it. "Light travels at the speed of light because it's a luxon" offers no explanation for why light travels at this speed, the term "luxon" is just the name we give to a particle travelling at that speed.

Seriously, @evan_au, you write very well and spend the time to put in references and hyperlinks. Most of the time they're the best and most easily read and understood things in every thread. People may read your posts, so in the future you just can't post if it gets too late.

The proof that 1+1=2 can not be simple arithmetic, because that would be circular reasoning.Circular isn't necessarily wrong. A proof can be one line if the statement you want to prove is an axiom.

As breifly hinted at above, Mathematics may be very circular. There may be many places you could start from to build up all the rest of Mathematics.

Is it meaningless?

Now, I wouldn't want anyone to think that Mathematics must be junk because anything based on a circular argument is usually junk. Quite the contrary, it may be one of the strengths of Mathematics and provide some assurance that it is something worth studying.

No matter what you start from and what you consider as your fundamental objects, you'll probably end up being able to re-create the entirity of Mathematics. So, for example, aliens on planet A who are more interested in some geometry that appears in their environment instead of sets should still end up with the same complete works of Mathematics that we have. Other aliens on planet B may be completely blind, they can't see any shapes, their main sense might be feeling vibrations (like a spider on a web). Studying the harmonics that can exist on a string will probably also allow them to develop the entirity of Mathematics, even though the natural number system they learn in school may seem very alien to us. For example a string vibrating with the first harmonic could be driven again at the same frequency and all that happens is that the amplitude gets bigger, the frequency doesn't change. So, for them, their logical number-like system is 1 + 1 = 1 If we said to them 1 + 1 = 2 they might think we were idiots who never went to school. However, their Mathematicians will realise that we are just describing a slightly different structure: If you join two strings together and each was vibrating with it's 1st harmonic, then you can end up with a new string (of twice the length) and that would be vibrating with its 2nd harmonic frequency, so they will understand that we are just using a different algebraic structure. Anyway, hopefully you get the point... all algebraic structures and all of mathematics could be re-created by studying the vibrations on a string.

Of course, this tacitly assumes that several things in nature do follow or have some structure which we may call an algebraic structure. Well.... it seems that it does but that's a discussion the scientists can have in another thread if they wish. Historically, mathematics was rooted in some stuff that seemed to appear in nature. Later on it's possible to just propose some new axiom system that may have no basis in nature or reason to assume its truth and still declare that you are perfoming or doing some Mathematics. However, I would still argue that this is justified by some structure existing or being exhibited in Nature. Specifically, we can study an arbitrary structure and axiom system because we can argue that there is some Mathematical system that does seem to hold in Nature, so it's sensible to conisder "what if" scenarios..... what if there were other axioms and other structures?

It's too late and too long already. I'm signing off.

Best Wishes.