Arithmetics and algebra, what/ which is the main concept?

1. Oct 7, 2011

logics

could you explain if we can define a main/basic concept of math and if terminology corresponds to concepts or is chiefly due to historical reasons:
set, set theory (ZF[C]) should be the founding concept, abstract algebra studies underlying sets, [elementary]arithmetics the result of logical operations between sets and math operations between their cardinalities/[dimensionless ?]numbers, elementary algebra operations between known and unknown numbers.

are numbers in arithmetics really dimensionless? is algebra just a variation of arthmetics or the latter a simplified form of algebra?

Last edited: Oct 7, 2011
2. Oct 7, 2011

chiro

Hey logics and welcome to the forums.

One of the staples of math is undoubtedly the idea of a set. It is no surprise that the operations of set theory, construction of sets, classification of sets, and axioms of set theory are also crucial as well.

The truth is that mathematicians, at least for a while now, try to clean things up, make things for formal, definitive, and rigorous and try to create generalizations in different ways. This has or is happening with many different fields that include calculus (ideas of measure theory, precise definitions of continuity, what is integrable in certain measures and other examples), axiomatic (or as "axiomatic" as we currently can do) (which involves set theory), as well as algebra (definitions of structures like groups, rings, and so on).

In terms of actual definitions, most are nowadays pretty clear, since many mathematicians have spent a great of time doing this precise kind of activity, and the first thing you start off learning is usually centered around sets.

In terms of quantity, then the most general number that most people use is the complex number. The truth is however that there are quite a wide range of esoteric formulations of quantities that obey all kinds of wacky rules and different kinds of algebras give specifics of this kind of behavior.

What I observe to be happening is that slowly we are going from looking at structures that are simply tensor products of complex numbers (like a vector or a matrix that has complex numbers in which the structures behave like you would expect them to), to structures where there are some more complex behaviors going on. If you want examples, take a look at the world of geometric algebras, which can get quite insane. If you want to get into theoretical physics, you'll probably have to end up understanding this kind of thing.

With regard to numbers being dimensionless, the reason for this is that you do not need to specify anything to obtain information about the structure. A point is a point and it is always constant so it has no dimension (dimensionless). A line however requires a parameter of some sort. It can be embedded in any dimensional space, but it is still one dimensional, and in order to get a point, you need to give it the parameter before you get that point.

The dimension of a system more or less is the number of free variables that you need to have to describe the object itself. Any infinite surface in Cartesian geometry is 2-dimensional because you need to describe the "up" component as well as the "right" component, even if it is something complex like a Riemann Surface (think sphere like the earth surface). For the "earth surface" you need latitude and longitude.

I'm not sure what you are asking though about the other stuff however.

3. Oct 7, 2011

logics

hallo Chiro, thanks for welcome:
numbers are obviously dimensionless, in physics we call dimension a concept expressed by [what Cantor calls] a sentential formula [, I do not know if this term is superseded] which identifies a set. A number is then considered for what it really is when there is a ratio between same dimensions.
what I mean is that when we do a sum the "dimension", the concept the elements of the set [which we are adding] share is "understood" or is undetermined, because we know we can add only apples and apples etc, so when in arithmetics we say 2+3=5 we, can and we, are in fact "eliding" the dimension and we mean : 2[apples, pears,] + 2 [apples,..] = 5 [...]
as to the other question, I was just wandering which term should have "primacy", but that is just a conceptual/theoretical curiosity. What I really would like to know is:

1) isn't there only one operation union/addition [and its reverse, when in reality it is possible]?
complex numbers, [as you righly say, are not numbers, and therefore] are not within ZFC,
2) is there any other exception?
3) can we say all math is ZFC? [btw: must I say ZF or ZFC, what is mainstream?]
now the big question:
4) they say Banach-Tarsky is a "paradox", why "that", why a surprise?, if we do not consider volume, then "that" sequitur from the property of form of being ubiquitous.

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4. Oct 7, 2011

chiro

Hey logics.

I'll start off by responding to your last question first:

There is a member here called micromass who has an interest in the Banach-Tarski Paradox, and he talks about it in his blog. Here is the link for his blog:

https://www.physicsforums.com/blog.php?b=2993 [Broken]

You can find other entries that talk about that, and hopefully he will come in here and give you some pointers.

In terms of really understanding dimension, you should start off by learning the concept of dimension in linear algebra. After that, you can then look at non-linear analysis and apply it to systems that are non-linear like Riemann manifolds.

With your pears and apples example, what you are saying is true, but it turns out that some-systems are not completely independent like the pear/apple scenario you talking about. In fact sometimes you can have things in high dimensions that turn out to be lower dimensional objects, like the line I was talking about in my first post.

If you want to understand this, imagine you have a physical piece of string. Now you could make it a straight line parallel to x or y axis, but you could also make it do all kinds of fancy loops and curves in three dimensional space or even higher dimensional space. The fact is that the object itself is one dimensional: you only need one parameter of variation to describe the object.

In terms of your "operation" question, in set theory we have two main operations which are union and intersection of sets.

In terms of dealing with numbers through, there all kinds of axiomatic systems that deal with different formulations of algebra, but that is a completely different discussion.

One thing that you should know is that the complex numbers were responsible for tightening the rules of arithmetic (+,-,*,/) since dealing with the square root of minus one makes things a little bit "weird".

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5. Oct 7, 2011

logics

I will not discuss your statements, I might if you wish, but please remember we are talking of elementary arithmetics and elementary algebra alone (and now complex numbers)
1) are complex numbers numbers?
if so what definition of number we need?
are they within ZF?
if they are not, are you aware of other exceptions?
2) at http://en.wikipedia.org/wiki/Quantity_calculus" [Broken]
the "calculus" of Dimensional analysis is defined as "analogous" to algebra. Is this statement correct? isn't just arithmetics?
3) what is , in your personal experience, the prevailing opinion: ZF or ZFC?
4) are there objections in principle to the reason of my acceptance of the paradox?
my questions are very simple

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6. Oct 7, 2011

chiro

No worries dude, but you need to really be more clear next time.

Complex numbers are indeed numbers. They have very specific axioms (like real or integer numbers).

Just to be clear, all of the complex numbers form a set and under some restrictions they form a group. If you want to understand algebra, I urge you to read about some group theory.

Also I think you are getting a little confused: ZFC is about set theory, not numbers. Sets contain "stuff"; the "stuff" is abstract: it could be real numbers, integer numbers, complex numbers, it could even be matrices or some really complex structure. Don't confuse set theory with other stuff.

This is just used in situations where you have quantities that take on some kind of physical form: which is why it is used in the natural science.

As an example velocity is just distance over time or /[t]. It doesn't get more complicated than that. Each quantity can either be atomic, or dependent on other quantities. For example time can be defined in terms of length, since it is the amount of time it takes light to go from here to the moon, or it could be measured in terms of an atomic process like the one cesium atom process measurement is based on.

If you want to know more about this: study a physics or an engineering textbook. The wiki page looks good, but it is really not that complicated.

I'm not an expert on foundational mathematics, and I don't devote much (if any) time to those kind of issues, but again I think you should ask micromass since he knows about the axiom of choice (HINT! HINT!)

As I recall micromass did say that in the Banach-Tarski situation, the measure that was being used was of measure zero, which raised an eyebrow for me.

Again I *URGE* you to look at the blog. You will find a lot of information that you are looking for.

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7. Oct 7, 2011

logics

thanks, "dude", that's more like it!, now, I see a contradiction there (1) about CN
1) if they have specific axioms, [as they surely do,] they are not [within] ZF.
What is the current definiton of number? does it include CN?
is utility of CN limited to solving the trivial problem of$\sqrt{}$-1?
ZFC "is" set theory and set theory is not about "stuff". It is the logical foundations of arithmetics [and math]. It shows how the operations in logic [with concepts] determine the operations in math [with numbers].
An element of a set is a concrete entity [countable noun] which posesses the property [measurable but not countable noun/adjective] that characterizes the set: if the concept is "beauty/beautiful" or "red" all the elements of that set are beautiful or red and "they" can be counted and summed.
2) it is a lot more complicated than that. Are you prepared to discuss QC and DA?, I made a thread that so far got no response
3) I should ask micromass?, how do you ask him? and [,if he ever replied] I can imagine the answer!, so I am glad you are taking my questions, thanks again!
4) I'll certainly look at the blog, but I am not looking for more information, I do not want to assert anything, or to discuss math. I am trying to get a clear idea of the situation:
math can only describe concepts, once you have defined your premises. You can make a false premise "pigs fly," and, from that, you can develop a huge and complex system of equations, matrices, claculus, tensors etc that can justify any conclusion. That can put off anyone who might be able to disprove the fallacy. I am only interested in concepts and implications

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8. Oct 8, 2011

logics

hi, Chiro,
I'd like to specify [in case somebody wants to respond] that I was referring to http://en.wikipedia.org/wiki/Banach-Tarsky_paradox" [Broken] : "...points that do not have a volume in the ordinary sense..."
if 'in priciple' we accept the idea that we may consider volume "not in the ordinary sense", regardless of formal proofs or technicalities, which I cannot judge, we may accept without wonder its consequences. The "paradoxex" are visible, self-evident in reality, in this world: matter is locked in one place, is "uniquitous"and cannot exist without form, form exists without matter, is ubiquitous and is dimensionless: 1) same sphere/ circle/ etc is [identical to another] anywhere/ everywhere at same time, 2) a sphere/ circle is [identical] as smaller or bigger one.
In addition, a sphere has a unique property even when we consider dimensions and volume in the "ordinary sense"

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9. Oct 8, 2011

pwsnafu

No, that is not obvious. There some mathematicians who consider complex numbers to be an example of a two dimensional number system. I don't agree with that view, and prefer the complex numbers form a two dimensional manifold, which is a subtle difference.

If we are talking about ZF, then "take the power set" of a set is also an operation.

You seem to have strange concept of "within". Complex numbers are constructed from reals, which are constructed from rationals, which are constructed from the integers, using the axioms of ZFC. Maybe not first order logic, I don't think Dedekind cuts are first order, but you never specified that.

No.
There are alternative formulations of set theory, and you have topos theory: a branch of category theory which studies "things that behave like sets". But there are plenty of topoi that aren't sets, and hence not covered by set theory.

Mathematics does not define what "number" means.
Any definition has long since replaced by set theory.

Analysists worry about axiom of choice because it leads to non-measurable sets.
Algebraists use Zeno without a second thought.
So the compromise is write "we use the axiom of choice".

It's the algebraic closure of the reals.
I don't know what else you want.

It's one possible foundation. It's also the one most used. But "is set theory"? No. Set theory is more than ZFC.

This screams of arrogance. You dismiss the opinions of one of the most experienced posters on the forum before he posted?

No reply. I just find it funny you come to a math forum and claim you don't want to discuss math in a thread discussing algebra and arithmetic.

10. Oct 8, 2011

logics

Hallo pwsnafu,
thank you for your warm welcome and for answering my questions. I hope you agree it is advisable to avoid:
1) "ad hominem argumenta" for three reasons at least, it is not elegant, two-can-play-that-game [if I were a child I could say that that screams arrogance, to begin with, and then say you are showing-off,...and conclude: "nomen omen"], and last but not least because this ancient, medieval informal fallacy is the meanest of them all, and was used by sophists who could not find better arguments. I can easily tolerate false assumptions, but I cannot tolerate, allow me:
false statements: (2) is brutally false: false, because I only said that it is impossible for me to ask/get anyone here to take my questions, brutally because you ignore that I already PM'ed twice the advisor, and I suppose that is humble enough of me. Then again, I hope you will not *deny me a mental faculty of/ and a right to/ imagining the probable [since I got no] answer. I'd appreciate that*. I find it [not funny] but not coherent/consequent with your post and your tone that you did not venture even an apology of a reply to the "paradox" question
3) that is blatantly false, too: this is the "logic" forum, not "general math" or "calculus&analysis" "linear&abstract algebra" forum, and I am making conceptual/theoretical/"logic" questions. I hope you got that far [my username is a broad hint, you know?]. "intelligenti pauca".

I'll discuss your answers one at a time, seriously, I will not be ironic any more. I do not expect an apology, but I hope you change your tune. I hope also you keep in mind that I repeatedly stated that I am limiting my compound to elementary arithmetics and algebra: ZF, if possible keep out complex number, we'll discuss it now and then forget . I'll put numbers to the issues, now:
0) I hope you could give a personal opinion on the OP: terminology and primacy
1) http://en.wikipedia.org/wiki/Number" [Broken] says "a number is a mathematical object used to count and measure...in math the definition has been extended to include zero, negative/rational/irrational [and complex CN] numbers". I a priori excluded CN because they do not obey the rules of arithmetics: *multiplication has its own peculiar definition. If you know or think that* compatible with ZF, [that is the meaning of "within ZF"], that that* can be explained/justified by set theory, please say so and argument that. Please read the explanation I gave of set theory above, and tell me if it is wrong or if you can give a better one. Please, stick to mainstream and do not tell me what "some" people think.
b) numbers: 3, -8, 3/4, 1.4142... are obviously dimensionless, pure. You question that?.. you sure the dimensions "some" people imagine are same dimension we are talking about?

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11. Oct 8, 2011

micromass

Staff Emeritus
Well, what is a number anyway?? I have rarely seen a definition of a number in mathematics, and I doubt that such a definition exists. But in any case, every definition of "number" must include the complex numbers. The complex numbers are way too important to not be considered number.

I don't know.

Absolutely!! The complex numbers can be well-defined in the theory of ZF.

ZFC, without any doubt. I'd say that 99% of all the mathematicians use ZFC. Rejecting the axiom of choice might be interesting because it might solve the Banach-Tarski paradox. But not having choice creates so many problems!! Not having choice would destroy analysis and algebra completely (or at least make it much uglier)

I would even say that most mathematicians use something stronger than ZFC. That is: they also accept another axiom that states that there is an inaccessible cardinal. This axiom is essential to make category theory work and for some cohomology theories. For example, the latest proof of Fermat's last theorem relies on the existence of an inaccessible cardinal.

Another new area of research is topos theory which claims to be a replacement for set theory.

But still, most mathematicians do accept choice. If you don't, then you'll quickly run into trouble!!

Your acceptance basiically says that we neglect the idea of volume, right?? Well, this is indeed a good approach. The modern acceptance of the axiom is indeed that not every set has a well-defined volume. Or at least: that the volume doesn't behave normally for these kinds of sets.

12. Oct 8, 2011

pwsnafu

Where did you say you did that?

I'll have to retract this one. I thought we were in Number Theory forum when I posted that.

Specifically what do you want me to address?
PS. What is primacy?

Huh? And multiplication of the real numbers has its own laws. As does fractions: it is certainly not "repeated addition". It would be nice if you defined "the rules of arithmetics".

The complex numbers are constructed using ZFC.
Therefore they are part of ZFC.
I honestly don't know why you have a problem with that.

Yep, I'll question that.
IF we are restricting the discussion to set theory and arithmetic, then dimension is not a property that has a definition.
We can talk about dimensions in linear algebra.
We can talk about dimensions in topology.
We can talk about dimensions in measure theory.
But not in arithmetic.
Similarly you can't talk about whether something is "dimensionless" either.

PS. I'll need a clarification. Do you want this discussion to only be about ZFC? You write

Last edited: Oct 9, 2011
13. Oct 10, 2011

logics

Thanks, micromass, for your replies, I studied set theory before that, when it was about "a property of ...algebraic numbers". [I hope, later on, you'll take a couple of questions on that, could you, for now, give me just link, please?]
I do not know how much time you can spend on this thread, so, I'd rather ask you in the first place:
I opened this thread because I had no reply in "the axiomatization of QC.." thread (question 3) https://www.physicsforums.com/showthread.php?t=536184". I think definition is clumsy, superficial [wiki can sometimes be, but probably this mistake has its roots in the vagueness of the concept of algebra, on one hand, and the erroneous definitions in DA ], because it equivocates a letter [(un)known quantity] substituting a number [3a+2x], with a letter substituting a "dimension" which, in its turn, is a misnomer of a concept identifying a set [2m*3v = 6m*v].
I believe the "dimension"/property in DA is just the understood/omitted [in arithmetics and algebra] "concept"/property identifying the set. Is that so? as I said above (post#3)
1+1 => [1 o $\in$ (Apple/Mass/Vector...) + 1 o $\in$(Apple/Mass/Vector...)] => 1+1= 2 [o $\in$ A] , 1+1 = 2 [ o $\in$ V] etc..
am I wrong?

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14. Oct 10, 2011

micromass

Staff Emeritus
A good reference is the book by Hrbacek and Jech: "introduction to set theory". I beloeve they construct the complex numbers...

15. Oct 10, 2011

SteveL27

Yes.

If you accept that real numbers are within ZF, then a complex number is just an ordered pair of real numbers. We can identify the complex number a + bi with the ordered pair of real numbers (a,b) with the appropriate multiplication.

So IF you believe the real numbers are part of ZF, then the complex numbers certainly are too; since ordered pairs can be defined within ZF.

Now, how do we know the real numbers can be defined within ZF? That's shown at the beginning of a course in Real Analysis. The reals can in fact be built out of the rationals. And the rationals can be built out of the integers; the integers can be built from the naturals; and the naturals can be built from the empty set!

It's a tower of magic tricks, but it's logically solid.

Now as far as "what is a number?" that's a good question. Consider the integers mod 5: that is, the numbers 0, 1, 2, 3, 4 with addition and multiplication mod 5. So 1+1 = 2, but 4+2 = 1 for example.

Are these numbers? Well, yes. But they're not the same kind of numbers as the reals or the integers.

Mathematicians have a very general concept of number. A number is anything mathematicians feel like calling a number if they can define some sort of addition and multiplication on it. That's not a formal definition. There is no actual definition of "number."

16. Oct 11, 2011

logics

thanks Steve, can you answer this question? it has been ignored at least three times. Are your explanations same as Hrbacek's?
I believe anything I read if it makes sense, it it is logical. Now, the current definition of multiplication is "repetition of addition" its result is the cardinality of a set with m sets with n elements each :[m = 2]* n=....[3 + 3], [-3 + -3], [3/2 + 3/2], [1.4142...+ 1.4142...] these are numbers, and the result of addition is same as multiplication. Is that true for CNs? if it is not: ( [a+bi]*[c+di]=[ac-bd]+[bc+ad]i ), can you briefly say how you fit them in ST?
As for definition: it is mandatory for a scientific theory to clearly define its terms: number is the main term od ST and math, if we do not give one/any definition, I suppose everything collapses or anything is possible and we step "through the looking glass". am I wrong?

P.S. can you suggest a symbol to express math logical equivalence: "rewrite as" and not the numerical value[6/4=3/2]: 6/4 => 3/2, [(3a+2b+a)*b] => [4ab+2b²]) etc, my home-made symbol => is clumsy and looks like $\geq$, $\supseteq$

Last edited: Oct 11, 2011
17. Oct 11, 2011

pwsnafu

Take two real numbers. For the purposes of this exercise I will pick e and ∏.
Convince me that the multiplication of the two is related to repeated addition.

We do define the "natural numbers". We also define "real numbers", "complex numbers", "hyperreals", "quaternions" and so forth. But not "number".

18. Oct 14, 2011

logics

I do not want to "convince" anyone, I would like a friendly discussion, to compare notes.

the numbers you mention: 3.14..., 2.78... are numbers like 3, -3, 2/3 [=0.666..], if we consider artithmetic and its operations. You can measure a rope and say it is 3.1415 m long. Whenever you use them concretely you round them up or down, if you sum or multiply them you get same result.

You cannot measure a yard with 1+2i. You can call it "number", but there is a relevant conceptual difference. Show me, please, how you fit this "number" in set theory, how it can express the cardinality of a set.
you'll oblige me if you answer questions in my previous post

19. Oct 14, 2011

pwsnafu

You didn't answer my question. I will ask you again: how does pi multiplied by e relate to repeated addition?
Arithmetic is not set theory.
Do you want to discuss arithmetic or set theory?

But you can't measure a rope and claim it is π m long.
You are talking about significant figures. That is not relevant to the discussion.
What? 2 plus 3 does not equal 2 times 3.
And you can't do it for a irrational number either. Or the hyperreals.
Heck you can't do it for 2/3 either.
Show how to express π as a cardinality of a set.
Look, the http://en.wikipedia.org/wiki/Cardinal_number" [Broken]are
$0,1,2,\ldots,n,\ldots,\aleph_0,\aleph_1,\ldots$
There is no π. There is no 2/3.
Just because it is not a cardinal doesn't mean it is not a number.
I did. Now will you answer questions in my previous post?

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20. Oct 14, 2011

logics

there were no questions in your previous post, sorry
If you want to discuss the conceptual difference between arithmetic and algebra, and the conceptual meaning of set theory you are welcome.
If you believe real numbers do not apply, leave them out, it is up to you. I am just saying that they fit in the wiki definition of number, while CNs do not.
could you tell me what is your definition of set theory, it is not Cantor's "On a characteristic property of all real algebraic numbers"?

Last edited: Oct 14, 2011