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Can I invent my own version of mathematics?

  1. Jan 12, 2013 #1
    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.
  2. jcsd
  3. Jan 12, 2013 #2
    "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.

    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 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.
  4. Jan 12, 2013 #3
  5. Jan 12, 2013 #4


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    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".

    Correct framework for what?

    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.
  6. Jan 12, 2013 #5
    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 ;) .
  7. Jan 12, 2013 #6
    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.
  8. Jan 13, 2013 #7
    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.
  9. Jan 13, 2013 #8


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    What about charges? They come as negative and positive, don't they?
  10. Jan 13, 2013 #9
    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
  11. Jan 13, 2013 #10


    Staff: Mentor

    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.

  12. Jan 13, 2013 #11
    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.


  13. Jan 13, 2013 #12


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    Staff: Mentor

    Charges are either negative or positive. What is physically impossible about the charge being -10 or +10?
  14. Jan 13, 2013 #13

    I like Serena

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    Homework Helper

    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.
  15. Jan 13, 2013 #14
    The mathematician says "If one more person enters the house, it will be empty".
  16. Jan 13, 2013 #15
    OK fine... Show me -$10.
  17. Jan 13, 2013 #16
    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.
  18. Jan 13, 2013 #17


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    Staff: Mentor

    Are you just trolling, or do you really not know what is a charge in the physical context?
  19. Jan 13, 2013 #18
    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!
  20. Jan 13, 2013 #19
    Ok. I'm still not sure what your "Show me -$10" comment was supposed to mean.
  21. Jan 13, 2013 #20
    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.
  22. Jan 13, 2013 #21
    Just emphasising (in relation to the "but bank charges can be negative" post) the fact that minus $10 is not actually a physical quantity.
  23. Jan 13, 2013 #22


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    Staff Emeritus
    Science Advisor

    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.
  24. Jan 13, 2013 #23
    Hahaha OK to be fair I just realised 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.
  25. Jan 13, 2013 #24
    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.

    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.
  26. Jan 13, 2013 #25
    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.
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