Can I invent my own version of mathematics?

[jobyts]

Say, I make some modifications to the basic assumptions and concepts to the existing Mathematics framework.

How would the Mathematics community decides if it makes sense or not.

Say, in my variation of Mathematics, I allow arithmetics between infinity. I understand infinity is a concept, not a number. But in my branch of Mathematics, an operation is not just limited to be used withing operands, but also between, say concepts.

Or, 0/0 is undefined by the current framework of Mathematics. It is undefined because 0/0 could assume any value. I'll apply the same logic and say, the root of a number is undefined since it could have more than one value. So in my version of Mathematics, i, the imaginary number does not exist. How does the Maths community decide which framework to go with?
How would the mathematicians convince me that having imaginary numbers is the correct framework.

My question is more generic on the acceptance of one framework from another. Not specific about the example modifications I was suggesting. Unlike science, Maths folks cannot ask for any evidence.

 

[Number Nine]

Unlike science, Maths folks cannot ask for any evidence.

"Maths folks" ask for proof. You can define whatever structure you want, and then you prove things about it. The complex numbers exist because someone defined a set that behaved a certain way and declared that it would be called the "complex numbers". This is perfectly fine, and is how most mathematical structures are created.

How would the mathematicians convince me that having imaginary numbers is the correct framework.

If you don't want imaginary numbers to exist, then don't work with the field of complex numbers; then they won't exist. You can define whatever you want, as long as it's logically consistent.

I'll apply the same logic and say, the root of a number is undefined since it could have more than one value

I can find a real number x such that x² = 4. Therefore, 4 has a square root in the real numbers. You can define some exotic "other" structure where square roots do not exist, but then you would no longer be working within the real numbers. This is perfectly fine, but square roots do exist in the real numbers; you don't get to claim that they don't.

 

[0xDEADBEEF]

Most of the stuff you can come up with has already been done. It just hasn't proven to be all that useful:
http://en.wikipedia.org/wiki/Real_projective_line
http://en.wikipedia.org/wiki/Affinely_extended_real_number_system

You set up your rules, you find out what follows. If the physicists can use it for something then it might become canon.

 

[HallsofIvy]

Or, 0/0 is undefined by the current framework of Mathematics. It is undefined because 0/0 could assume any value. I'll apply the same logic and say, the root of a number is undefined since it could have more than one value. So in my version of Mathematics, i, the imaginary number does not exist. How does the Maths community decide which framework to go with?

You could do that but there is a difference between saying "could assume any value" with no reason to choose one above the other and say "could assume two values" where one is positive and the other negative. We can, and do, simply define 'square root of a" as "the positive number whose square is a".

How would the mathematicians convince me that having imaginary numbers is the correct framework.

Correct framework for what?

My question is more generic on the acceptance of one framework from another. Not specific about the example modifications I was suggesting. Unlike science, Maths folks cannot ask for any evidence.

No, but they can ask for "proof". In philosophy terms, sciences defines "truth" in terms of congruence with reality (a fundamental concept of "realist" philosophies) while mathematics defines "truth" in terms of consistency (a fundamental concept of "idealist" philosophies).

Yes, you can, and mathematicians do, set up whatever "axiomatic systems" you want. The question, then, is do you get useful (whether to scientists or mathematicians) and interesting results.

 

[BenG549]

So in my version of Mathematics, i, the imaginary number does not exist. How does the Maths community decide which framework to go with? How would the mathematicians convince me that having imaginary numbers is the correct framework.

Someone above already made this point but the end point is application of your ideas, in the case of imaginary numbers we can look at another example i.e. negative numbers. Negative number don't actually exist per se, you can't actually have negative something, it is a physically impossible quantity, you can't physically have negative 1 banana. However in the way we model the physical world the idea of negatives is useful and when we use them in mathematics our results reflect the what we observe in reality i.e. you have 1 banana if someone takes it you have 1 banana -1 banana and hence have 0 bananas. Imaginary numbers, like negative numbers, do not physically exist but are a tool, they help us understand and explain various process (without going into too much detail) we use them a lot in signal processing for example and the obtained results reflect the observed behaviours we are trying to model. Again someone has made this point above but when the ideas we are trying to convey have no obvious physical application that can be used (nothing to see if your results are physically right) logical constancy is the only measure of whether your new maths is reasonable, that is how mathematicians will test your ideas... Good luck with it ;) .

 

[Bill Simpson]

If I remember correctly then "Nonplussed!: Mathematical Proof of Implausible Ideas" by Julian Havil was a relatively introductory book that let students explore "what if" in math. If you can find a copy in your library or buy a copy from somewhere then it might give you a start in looking at alternative ideas in math.

 

[HomogenousCow]

Well, I guess you could come up with some mathematical object, and define some operations between them.
Most of the objects we study in physics, like vectors and tensors are just elements of linear vector spaces imagined by mathematicians.

 

[Borek]

Negative number don't actually exist per se, you can't actually have negative something, it is a physically impossible quantity

What about charges? They come as negative and positive, don't they?

 

[GarageDweller]

Someone above already made this point but the end point is application of your ideas, in the case of imaginary numbers we can look at another example i.e. negative numbers. Negative number don't actually exist per se, you can't actually have negative something, it is a physically impossible quantity, you can't physically have negative 1 banana. However in the way we model the physical world the idea of negatives is useful and when we use them in mathematics our results reflect the what we observe in reality i.e. you have 1 banana if someone takes it you have 1 banana -1 banana and hence have 0 bananas. Imaginary numbers, like negative numbers, do not physically exist but are a tool, they help us understand and explain various process (without going into too much detail) we use them a lot in signal processing for example and the obtained results reflect the observed behaviours we are trying to model. Again someone has made this point above but when the ideas we are trying to convey have no obvious physical application that can be used (nothing to see if your results are physically right) logical constancy is the only measure of whether your new maths is reasonable, that is how mathematicians will test your ideas... Good luck with it ;) .

Charge can be negative, and this is unavoidable, when you bring two point charges of opposite charge infitesimally close, you end up with no field, and hence no charge

 

[Mark44]

Negative number don't actually exist per se, you can't actually have negative something

Tell that to my bank. If I write a check for more than I have in my account, my bank charges me an NSF (nonsufficient funds) fee.

, it is a physically impossible quantity, you can't physically have negative 1 banana.

What about charges? They come as negative and positive, don't they?

 

[BenG549]

What about charges? They come as negative and positive, don't they?

Yeah obviously charges can positive and negative, but that is conceptual... you don't actually have negative money, it's just a way we describe our observations. Negative something is not a physical quantity is it? When you are charged $10 you aren't given -$10. But it's how we conceptualise that fact you have $10 less than before the charge.

Negative numbers were not even understood or used (widely in England anyway) until around the 1700s and there was a consensus then that they represented non physical quantities and didn't 'actually exist'*. Even in their earliest from they are a far newer idea than the concept addition of real numbers, for obvious reasons. You can count, see and observe a positive number of things, but you can't actually count physical objects to less than nothing. The first known number systems are thought to be used to track the passing of time and this dates back around 3000 years before the first record of negative numbers, even base 10 systems were in use thousands of years before people even considered that a negative number could in fact exist in mathematics**. Again they are used now as a way of conceptualising what we observe i.e. I have $10 less than I did yesterday... I can't have negative $10, and if that isn't true, then you can show me negative $10.

*http://nrich.maths.org/5961

**http://en.wikipedia.org/wiki/Number

 

[Borek]

Yeah obviously charges can positive and negative, but that is conceptual... (...) Negative something is not a physical quantity is it?

Charges are either negative or positive. What is physically impossible about the charge being -10 or +10?

 

[I like Serena]

A quantum physicist, a biologist, and a mathematician are sitting in a bar, observing an empty house across the street.
Two people go into the house.
A while later 3 people emerge.

The quantum physicist says that there was indeed a chance that someone materialized in the house by chance.
The biologist says congratulations, because a baby was born.
The mathematician says that there are now minus one people in the house.

 

[Number Nine]

A quantum physicist, a biologist, and a mathematician are sitting in a bar, observing an empty house across the street.
Two people go into the house.
A while later 3 people emerge.

The quantum physicist says that there was indeed a chance that someone materialized in the house by chance.
The biologist says congratulations, because a baby was born.
The mathematician says that [STRIKE]there are now minus one people in the house[/STRIKE].

The mathematician says "If one more person enters the house, it will be empty".

 

[BenG549]

Charges are either negative or positive. What is physically impossible about the charge being -10 or +10?

OK fine... Show me -$10.

 

[Number Nine]

OK fine... Show me -$10.

Er...why? Why does that have to do with anything?
Show me an irrational number of dollars. You can't, therefore, irrational numbers don't exist.

 

[Borek]

OK fine... Show me -$10.

Are you just trolling, or do you really not know what is a charge in the physical context?

 

[BenG549]

Er...why? Why does that have to do with anything?
Show me an irrational number of dollars. You can't, therefore, irrational numbers don't exist.

Is EXACTLY my point!

They are conceptual tools that when used in a consistently logical way can be used in mathematics to describe and model our observations, we know they are right (or at least useful) not because we can see and touch them, as they are not physical, but because they are logical and provide with data that, to an acceptable degree of accuracy, reflect what we observe!

 

[Number Nine]

Is EXACTLY my point!

They are conceptual tools that when used in a consistently logical way can be used in mathematics to describe and model our observations, we know they are right (or at least useful) not because we can see and touch them, as they are not physical, but because they are logical and provide with data that, to an acceptable degree of accuracy, reflect what we observe!

Ok. I'm still not sure what your "Show me -$10" comment was supposed to mean.

 

[BenG549]

Are you just trolling, or do you really not know what is a charge in the physical context?

lol, to be honest I am starting to think this is some kind of wind up... the bit about negative numbers in my original post was not really part of the point I was trying to make, but I am struggling to believe the that people can't accept that it is PHYSICALLY impossible to actually posses negative anything. I've given links to examples of mathematicians who have had this same discussion and agreed that it was a non physical quantity and saying bank charges over and over has nothing to do with it, in fact that is my point... they don't actually give you -$10 do they? you don't physically have -$10, but we can describe what we see using the idea of negatives.

 

[BenG549]

Ok. I'm still not sure what your "Show me -$10" comment was supposed to mean.

Just emphasising (in relation to the "but bank charges can be negative" post) the fact that minus $10 is not actually a physical quantity.

 

[Ryan_m_b]

This thread is drifting off course. I feel it would also be worth members defining negativity in the context of the conversation. Negative charge is qualitatively different than negative numbers and there may be some confusion in how different people are using the term.

Either way we should get back to the OPs point.

 

[BenG549]

Are you just trolling, or do you really not know what is a charge in the physical context?

Hahaha OK to be fair I just realized the mistake I was making with this. I had it fixed in mind that you were referring to bank charges and 'physicals objects' as opposed to electrical charges and physics for instance. Your still just using negative in a conceptual sense to describe something physical.

Plus positive and negative charge (in electromagnetism) is obviously not used in the same way as is in mathematics. Having excess electrons in an object... i.e. being negatively charged, obviously has nothing to do with negative numbers in maths (which is what the post is about)... hence my confusion.

Makes me wonder why you brought up electrical charge actually, but I guess that's even further off topic.

 

[VantagePoint72]

Plus positive and negative charge (in electromagnetism) is obviously not used in the same way as is in mathematics. Having excess electrons in an object...

The charge of the electron is -e. The charge of the positron is +e. It has nothing to do with excess or deficit: physical charge comes in two fundamentally different varieties, positive and negative. Which ones we call "positive" and which we call "negative" is arbitrary, but describing both at the same time in the same consistent framework requires both a "positive something" and a "negative something" exist. The fact that you can increase the net negative charge by either incrementing the number of negative charges or decrementing the number of positive charges (and likewise for a net positive charge) is in direct correspondence with the fact that you can get make a number's "negativeness" increase by adding -1 or subtracting +1.

i.e. being negatively charged, obviously has nothing to do with negative numbers in maths (which is what the post is about)... hence my confusion.

It's pretty clear that no one can make heads or tails out of your arbitrary reasons for deciding that the relevance to one set of numbers (the naturals) to one particular aspect of reality (discrete quantity) is sufficient to say those numbers "really exist", while the exactly analogous relevance of another set of numbers (the integers) to another particular aspect of reality (charge) is not. But I will certainly agree that you seem confused.

 

[VantagePoint72]

Either way we should get back to the OPs point.

Fair enough, though it seems like OP's question has been pretty thoroughly answered too so there doesn't seem to be much to get back to.

 

[homeomorphic]

Fair enough, though it seems like OP's question has been pretty thoroughly answered too so there doesn't seem to be much to get back to.

I beg to differ. In particular, with regard to imaginary numbers. Of course, it's correct, in principle, to say that you could always write down whatever assumptions you want and then see what you can prove from them. In practice, though, that's not the way it works. That wasn't how imaginary numbers came to be accepted. In fact, mathematicians, including the very mathematicians who INVENTED imaginary numbers viewed them with great skepticism for a long time. Two things happened. One was that it was found that imaginary numbers were involved in solving cubic equations. Not just for producing imaginary solutions, but for obtaining the real solutions in some cases. In this case, Viete was eventually able to solve the problem without using imaginary numbers, but it was an interesting curiosity that these numbers, which seemed to be nonsensical could actually be used to solve problems involving real numbers.

The second thing that happened was that people developed a way to visualize complex numbers and how they multiply on a two-dimensional plane. From that point of view, complex numbers are very natural because they are a way of describe rotations and dilations of the plane--the fundamental symmetries of Euclidean plane geometry.

 

[BenG549]

It's pretty clear that no one can make heads or tails out of your arbitrary reasons for deciding that the relevance to one set of numbers (the naturals) to one particular aspect of reality (discrete quantity) is sufficient to say those numbers "really exist", while the exactly analogous relevance of another set of numbers (the integers) to another particular aspect of reality (charge) is not. But I will certainly agree that you seem confused.

Well I've posted links to examples of mathematicians reaching a consensus about the fact negative quantities are 'not real' as far back at the mid 1700s.

Arbitrary reasons? I would probably say the fact that I have never seen an example of a negative physical quantity makes my assertion, that it is a conceptual way of modelling our observations, reasonable. If I am wrong then feel free to show me an example of being able to have less than nothing of a physical quantity?

Adding electrons to an object and increasing is negative charge is not a concept lost on me but my point is that when you remove electrons you are not giving an object minus n electrons (less than zero electrons), but mathematically we can describe increasing positive, or reducing a negative charge by subtracting electrons i.e. adding (-n electrons).

You can't honestly say, in the world as we know it, that you can have possession of a negative number of things... as far as I am aware that is physically impossible, that is my only point. Is that not reasonable?

 

[BenG549]

The second thing that happened was that people developed a way to visualize complex numbers and how they multiply on a two-dimensional plane. From that point of view, complex numbers are very natural because they are a way of describe rotations and dilations of the plane--the fundamental symmetries of Euclidean plane geometry.

Exactly, they are simply used as a means of describing physical things (oscillations or describing any time varying signal being the obvious ones for me as an Acoustics student)

I think the fact that hey are described as 'Imaginary' causes a bit of a problem for people not using them in a practical setting, René Descartes, wrote about them in his La Géométrie, where the term imaginary was first used (as far as I'm aware) and it was simply meant to be derogatory due to the lack of application at the time.

 

[VantagePoint72]

Arbitrary reasons? I would probably say the fact that I have never seen an example of a negative physical quantity makes my assertion, that it is a conceptual way of modelling our observations, reasonable. If I am wrong then feel free to show me an example to being able to have less than nothing of a physical quantity?

You've decided that the only "real" numbers are those that correspond to quantities of objects, and numbers that correspond to things other than quantity are just useful fictions. You are being given concrete examples of attributes that are naturally identified with the negative numbers, but arbitrarily rejecting them because they're not negative quantities. You're so busy demanding someone show you a negative quantity—which no one is claiming to do—that you're failing to see that what is being claimed by pretty much everyone you're arguing with is that numbers can correspond to other physical properties that are just as meaningful as quantity. If you're going to argue that one set of numbers is somehow a "real thing" because can be put into a direct correspondence with some physical property, then rejecting other numbers as "real things" just because they're in direct correspondence with some other physical property is arbitrary and unjustifiable.

You have said, "My point is..." or some variant quite a few times now. You might consider the possibility that you're not being misunderstood by everyone, but are the one not understanding the point being made.

If you have some argument in favour of discrete quantity being somehow more a fundamental property of reality than every other and deserving of special treatment, then by all means—but if your reply contains a sentence like, "Show me a negative amount of ________," then you may as well not bother. If that's your reply (again) then you don't understand the arguments being made against your position.

 

[BenG549]

You've decided you're failing to see that what is being claimed by pretty much everyone you're arguing with is that numbers can correspond to other physical properties that are just as meaningful as quantity.

Hahaha, Well obviously that is true but if that is the case I have NO idea why everyone started getting on my back about this! I never said describing other physical qualities was any less meaningful... quite the opposite

In my original post, as an example of how logical reasoning defines whether your maths is right, I said 'in the way we model the physical world the idea of negatives is incredible useful and when we use them in mathematics our results reflect the what we observe in reality', which is evidence of the fact they were developed though reasonable logic.

I never said that negative numbers are useless because they're not real.

You have said, "My point is..." or some variant quite a few times now. You might consider the possibility that you're not being misunderstood by everyone, but are the one not understanding the point being made.

Oh the thought definitely crossed my mind lol, and I have asked people out side of this little discussion if I'm being unreasonable, but maybe they are just as stubborn or ignorant as me.

 

[BenG549]

Did anyone still arguing with me actually read my original post?

 

[chiro]

For the people that say irrational numbers don't exist:

We could if we wanted to, designate that the standard base is SQRT(2) instead of 10 which means representing 10 in this case would end up being irrational and SQRT(2) would be rational in this base.

In fact, all languages have this property: In some bases you have an optimal language for an optimal representation and you also everything else that is not optimal.

In another language what was un-optimal in the other is optimal in this one and what was optimal in the other is un-optimal in this one.

Remember that the numbers themselves are just labels in some descriptive capacity and that descriptive capacity has advantages, disadvantages and limits to not only what it can describe, but more importantly how.

Many people forget that integer bases are not the only ones that are used for counting and for representation of numeric quantities.

 

[VantagePoint72]

I never said that negative numbers are useless because they're not real.

Sigh. No, but you're still saying they're not "real" (as in, don't really exist), while arguing that natural numbers are because they correspond to quantity. That's the inconsistency. I'm fully aware that you're not saying non-natural numbers are useless and I have you idea how you got the impression that's what I was saying.

Did anyone still arguing with me actually read my original post?

Yes. The original post that said:

Negative number don't actually exist per se, you can't actually have negative something, it is a physically impossible quantity, you can't physically have negative 1 banana.

If you genuinely don't see that what everything that has been said to you by various people is a direct response to this claim, then I doubt repeating it yet again is going to do it.

 

[BenG549]

Sigh. No, but you're still saying they're not "real" (as in, don't really exist), while arguing that natural numbers are because they correspond to quantity.

Yeah OK 'real' was a VERY bad choice of word on my part (especially given we are discussing mathematics), I guess I didn't mean real because that might suggest that I feel other mathematical ideas are fictional or that negative numbers do not exist on a number line and so on. I'm not saying irrational numbers etc don't exist... I've used them, I know they exist, but in the case of irrational numbers they are uncountable (Cantor's diagonal argument?.. someone with a maths degree might be able to qualify that) you can't observe an infinite series of digits or sets that is what I meant by not real (I accept that it was a bad word to use but in the context of my opening post I didn't think it was going to lead to this lol).

If you genuinely don't see that what everything that has been said to you by various people is a direct response to this claim, then I doubt repeating it yet again is going to do it.

Yeah, you're probably right. Thanks for trying anyway. Didn't mean to annoy anyone too much. I guess I should just concede that I'm either being wildly miss understood or too stupid to understand why I'm wrong, either way I should probably give up.

 

[BenG549]

Remember that the numbers themselves are just labels in some descriptive capacity and that descriptive capacity has advantages, disadvantages and limits to not only what it can describe, but more importantly how. Many people forget that integer bases are not the only ones that are used for counting and for representation of numeric quantities.

Exactly! This is what I've been trying to say.