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Is Mathematics an ideology?

  1. Jul 26, 2015 #1
    Here is a quote from a person from another forum (he's not replying to me. I am just a lurker) But since I don't know much about mathematics. I wonder if he's right. Is mathematics an ideology just like Democracy, Communism, etc...?

    Here is a quote from a person from another forum (he's not replying to me. I am just a lurker) But since I don't know much about mathematics. I wonder if he's right. Is mathematics an ideology just like [edited] political ideologies.
    All of this argument was started by the concept of a better ideology. If Mathematics has assumptions and be correct - then so can other ideologies". [

    Another point was since a lot of higher mathematics proofs and equation have no physical or empirical evidence, then that makes mathematics ideology.

    "Not at all, since these rights are used as defining parameters to democracy. They are no different than defining parameters in various other systems that i have given examples of."
    "If other systems can have defining parameters, so can a political system."
     
    Last edited: Jul 26, 2015
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  3. Jul 26, 2015 #2

    Drakkith

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    A word of warning to anyone posting, please limit your discussion to the mathematical part of this and refrain from going off on a tangent about any political/philosophical aspects which may/may not be correct or that are opinion based.

    The original question is: "Is mathematics an ideology just like Democracy, Communism, etc...?"
     
  4. Jul 26, 2015 #3
    To the person who posted this, please do explain why. I know it sounds stupid, but please. Thank you.
     
  5. Jul 26, 2015 #4

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    There are very fundamental characteristics of mathematics that distinguish it from an ideology:

    1) The basic assumptions of mathematics were slowly developed from the need to do basic counting and calculations. A study of abstract algebra or plane geometry show that the assumptions are very basic and natural. There is nothing mythical about them and, in fact, it would be difficult to imagine alternatives. (Many alternatives mentioned in the OP for a different number system would not really do that. They would only change the names or symbols used in mathematics.)

    2) Mathematics is used all the time to make predictions that are verified to be true. Every time a calculation of the area of a square is compared with the measured result, the calculation turns out to be true. So math has been used to predict results trillions of times without a counterexample.

    3) Mathematics reacts to ANY counterexample that is discovered. And the reaction can take the form of a change to its very foundational assumptions. In pure math, the assumptions are not contradicted by any known examples. Even the most basic assumptions are open to question and revision if a counterexample is found.

    These are not characteristics of an ideology.
     
    Last edited: Jul 26, 2015
  6. Jul 26, 2015 #5

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    That depends on your definition of addition. You can define addition so that 1+1=2 when looked at in one setting, but not in other settings. Suppose addition is defined as adding measurements around the Equator, and 1 = the distance half way around the Earth. Then 1 + 1 gets you all the way around the Earth's Equator and back to where you started. So 1 + 1 = 0. In that abstract algebra, there is no number 2.
     
  7. Jul 26, 2015 #6
    What about this part?
     
  8. Jul 26, 2015 #7

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    If you are interested in fundamental questions like this, you might be interested in abstract algebra. Suppose you define '1' as something and '+' is defined. Then '1+1=2' says that '2' is the result of '+'ing '1' with itself. Addition might represent adding distances end to end in a straight line on a plane. But it might represent something else entirely (rotations of an object, chemical reactions, etc.). But in all cases, the symbol '1' represents something and '1+1' means that you are adding two of the same thing. If the numbers represent distances, that translates to adding two of the same distance together. The result is denoted by the symbol '2'.
     
  9. Jul 26, 2015 #8
    I haven't read much of this, I didn't get beyond "That is because everyone assumes and agrees that two whole numbers should be evenly spaced out. There is no empirical basis to that" which is the exact opposite of the truth.

    "1" is not an invented concept, it is something that we observe. When I look at you, I can see 1 person. If I have a collection of similar objects in front of me I can divide them into groups. If I keep dividing the groups until I can't divide them any more, each group will contain exactly 1 object.

    If I pick up 1 of the objects, I have 1 object in my hand. If I pick up 1 more object I have 1 more than 1 object in my hand, and we assign the label "2" to the number of objects in my hand. If I pick up 1 more object then I have 1 more than 2 objects in my hand and we assign the label "3" to the number of objects, and so on. In this way we can observe that whole numbers are evenly spaced out, it is not an assumption.

    Edit: I don't know about the "other forum" but I'm pretty sure that reposting wholesale from PhysicsForums without permission is a breach of copyright, so if you were thinking of doing that, don't.
     
    Last edited: Jul 26, 2015
  10. Jul 26, 2015 #9

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    I think that the short answer is that the equal spacing is true because of the definitions.
    Suppose '1' represents a distance, and '1' + '1' represents the sum of two equal distances '1' placed end-to-end in a straight line on a plane. That distance is represented by the symbol '2'. By those definitions, '2' is always twice the distance of '1'.

    There can be other definitions in other geometries where distance is not additive. You can 'add' distances around a circle and end up with a zero total distance.
     
  11. Jul 26, 2015 #10
    I lurk and I don't post there. So no worries about copying your posts. I just wanted to understand what they mean and I am not sure what they mean and what better than you guys, I thought.
     
  12. Jul 26, 2015 #11

    Nidum

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    Is it not true that whilst mathematics can be based on any arbitrary number system and methology whatever is actually being described, quantified or analysed is actually always the same thing deep down ? Therefore use whatever number system and methodology you like but any description , quantification or analysis can always be mapped back into the conventional simple system ?

    Most mathematics is axiom based . Mathematics based on axioms chosen is seen to work . Axioms are shown to be valid .

    Man has always sought to understand and quantify his environment and has always striven to understand and quantify in the simplest way .
     
  13. Jul 26, 2015 #12
    No. The mathematics of whole numbers is based on (an abstraction of) the observation I described, which is clearly invariant and not arbitrary.

    Axioms are not "shown to be valid": they are the things which we ASSUME to be valid in order to show that other statements are valid.
     
  14. Jul 26, 2015 #13

    HallsofIvy

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    That's not the point. You did copy a great deal fro the other forum and post it here! Did you have permission to do that?

    As far as "ideology" is concerned, mathematics is not an "ideology" because no mathematician is saying "this is true" but rather "if this is true then ...". All mathematics statements, even if they are not written in that form.
     
  15. Jul 26, 2015 #14

    Ssnow

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    No, I think it is not an ideology. In front of an idea you can believe it or not and you should rely only on feelings. In front of a theorem you can know whether it is true or false.
     
  16. Jul 26, 2015 #15
    I just read the rules since I lurk. Yes, it is fine. I don't see anything saying no as long as you don't use it as your own work.

    Yea, I lurk so I don't really pay attention to any of it.

    I'll post the original link of the history forum, but I avoid it in case it provokes a thread close-down since it involves politics as a mod here warned me. No politics on math forums. Don't read it so this thread doesn't get entangled with politics. I'm just curious about the explanation of the math.
    http://historum.com/asian-history/92795-who-would-you-have-supported-1962-sino-indian-war.html
    I was also embarassed at my leap from a non-science history forum to a hard math one. It would probably make posters here have a good guffaw and stop posting serious responses, so please ignore it.
     
    Last edited: Jul 26, 2015
  17. Jul 26, 2015 #16

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    A lot of what is in the original post that you copied is mistaken. 2 - 1 = 3 - 2 is true because of how the symbols of 1, 2, 3 and + are defined. They are not assumptions.
     
  18. Jul 26, 2015 #17

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    It ain't no capital crime, but you should avoid it. There are lawyers everywhere. And it is bad manners in any case.
     
  19. Jul 26, 2015 #18

    Part of the OP's quote
    Isn't the linear system of numbers an assumption and that's what he's attacking here?
     
  20. Jul 26, 2015 #19

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    The number system is an algebra with an operation '+'. It can be used in many ways. Some applications have a 'linear' meaning on a straight line. Others do not. Here is (roughly from vague memory) a way to start building up a number system and it's properties that is not based on ideology:

    Suppose we have a set of elements and an operation '+':between any two elements
    1) If there is one element which leaves all elements unchanged when it is added, we will denote it by '0'. So e+0 = e for all elements e
    2) If each element, e, has another element that, when added, gives 0, we will denote that by -e. So e+(-e) = 0 for all elements e.
    3) If there is one element which can generate all other elements by repeatedly adding or subtracting it, we will denote it by '1'. So 1, -1, 1+1, -1-1. 1+1+1, .-1-1-1,... are all the elements.
    4) If two things are equal, and the same operation is performed on both, then the results are equal.

    With those conditions satisfied, lets define the symbols '2'=1+1; '3'=1+1+1; ...
    Then a lot of what your quote questions is forced to be true. It is not 'ideology'.

    A lot of what your quote says is assumed as ideology is really not assumed. They are postulated. There is a difference. When one studies Euclidean geometry, some things are postulated and certain results are obtained. But those things are not postulated in non-Euclidean geometry, and the results are different.

    This is the field of abstract algebra and it is very well developed and formalized. I am afraid that I am not doing it justice, working from memory.
     
    Last edited: Jul 26, 2015
  21. Jul 26, 2015 #20

    The definition of postulate (from Google) is "suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief."

    It sounds like an assumption, if not what is the difference in mathematics?

    But if you postulate A, and get Aa result and you postulate B and get Bb result - what exactly does this mean? Doesn't this mean the mathematics are not true and universally applied?
     
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