What is a negative number? : Philosophy Forums

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What is a negative number?

jmafoko
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Posted 01/12/09 - 02:08 PM: Subject: What is a negative number?	#1
I have seen definitions of negative numbers as equivalent classes of ordered pairs (a,b) where multiplication is defined as (a1, b1) x (a2, b2) = (a1 x a2 + b1 x b2, a1 x b2 + b1 x a2). Where does this magical formula come from? Out of the blue? It is this formalist's hat tricks that lives me uneasy. The problem is that it is too dry. An alternative approach(intuitionist or constructionist approach) to explain the above was given by Descartes. He says given "false roots" x1, x2 (negative numbers) and their corresponding equations, x1 +1 = 0 and x2 + 1=0, it can be shown x1.x2 = +1. He defines a false root as one whose sum with unity vanishes. Mr Descartes arguments depends on analytic/synthetic reasoning distinctions. If I understand him well, by an analytic object he meant one that can only be known only indirectly through its effects: ie first assume it exists and assume that it is multipliable just like all other numbers we know, and deduce its effect(like we did above). Some people began having problems with these mystery objects in a black box(negative, imaginary numbers, the in-divisible,), and demanded a mathematical theory. Nowadays there are better definitions based on abstract algebra. It seems to me that this whole discussion revolves around the debate between the constructionist and formalists school of thought. Is the formalist really conjuring up objects of his mind and then deducting theorems from them; are these new objects really what we intuitively feel about negative numbers; are there negative numbers out there and a mathematical model of the formalist down here; and if so, how can we be sure they match? For these reasons, the arbitrariness of the formalist approach to mathematics still unsettles me. This is not only a problem for negative numbers, I have the same problem with definitions of Riemann's integrals, differentials. line integrals, multi-integrals, dot products, cross products, etc.

rabeldin
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Posted 01/12/09 - 02:25 PM:	#2
The essence of mathematics is to define new things and derive how they work and see how they extend existing notions.
Leave no assumption unquestioned.

Dragohunter
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Posted 01/12/09 - 02:50 PM:	#3
I agree with rabeldin. I think mathematics is merely a language to describe how things work in the universe not the actual thing of it.
"Technological progress is like an axe in the hands of a pathological criminal." - Albert Einstein Influences: Ludwig Wittgenstein, C.S. Peirce, Soren Kierkegaard, John Searle, Socrates, Bertrand Russell, Jesus, Karl Popper, David Hume, Locke, Albert Schweitzer, Bernard Williams, Nietzsche, Richard Feynman, Zhuangzi, Albert Einstein, Leo Tolstoy

ManiacJack
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Posted 01/13/09 - 10:46 AM:	#4
Negative numbers are spatially contrary to positive numbers. They are the red and blue of the color spectrum, yet so many colors sit on the color wheel. Oh, how I wish Descartes preferred the Wheel to the Line. He was so very much against 'imaginary numbers', which of course is what he was running away from with the Cartesian Plane; the circle with radius of infinity.
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jmafoko
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Posted 01/13/09 - 12:34 PM:	#5
@dragonhunter "I agree with rabeldin. I think mathematics is merely a language to describe how things work in the universe not the actual thing of it." Can you kindly give an example of such things in the world. And how -1x-1=+1 can be mapped(interpreted if you may) onto them.

jmafoko
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Posted 01/13/09 - 12:37 PM:	#6
@Maniac Not sure if i get your point, especially the wheel metaphor. Please elaborate. @everyone Thanks for your responses.

Dragohunter
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Posted 01/13/09 - 03:04 PM:	#7
jmafoko wrote: @dragonhunter "I agree with rabeldin. I think mathematics is merely a language to describe how things work in the universe not the actual thing of it." Can you kindly give an example of such things in the world. And how -1x-1=+1 can be mapped(interpreted if you may) onto them. Can you give me an example of a negative distance how you say that can be mapped?
"Technological progress is like an axe in the hands of a pathological criminal." - Albert Einstein Influences: Ludwig Wittgenstein, C.S. Peirce, Soren Kierkegaard, John Searle, Socrates, Bertrand Russell, Jesus, Karl Popper, David Hume, Locke, Albert Schweitzer, Bernard Williams, Nietzsche, Richard Feynman, Zhuangzi, Albert Einstein, Leo Tolstoy

ManiacJack
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1 of 1 people found this post helpful Posted 01/13/09 - 03:30 PM:	#8
imafoko, Negatives are a direction inferred from the usage of displacement and not distance, as we all know distance is absolute. Displacement can be any which way we wish it to be, as all we need to do is add an adjective to a number line. Consider the rules: i * i = -1 i * i * i = -i -1 * -1 = 1 -1 * -1 * -1 = -1 1 * 1 = 1^2 1 * 1 * 1 = 1^3 What is funny is that there appears to be no pattern to the results[right of equals sign], though there is to the inputs[left of equals sign]. So we are missing some basic rules that apply to our understanding of algebra. Firstly, that relativity is missed, obviously, because of the apparent incongruousness of values- that is, -1 and 1 follow different rules than each other in regards to identity, though all are displacements from the origin. Additionally, the cartesian plane recognizes only two-way symmetry; I think there are many more, ala the Wheel; the Unit Circle; the Basis for Geometry; the metaphorical One. The difficulty with knowing a similitude of negatives and positives lies in the prescription that they are polar opposites and exhaustive in that regard. Again, what of Imaginary Numbers? They are necessary, yet we do not say they are opposite of Negative Numbers! It is because of this that the Trifecta is so hated; it buries the law of non-contradiction; it destroys the absoluteness of the equality sign. It rends relativity, not absoluteness; it rends geometry, not algebra. So, you won't find your answer within that domain of Algebra. Algebra is designed to make the negative and positive and imaginary as senseless to one another- to keep them unknowable. Human Error. I figure rediscovery is the solution. Edited by ManiacJack on 01/13/09 - 05:37 PM
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jmafoko
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Posted 02/10/09 - 03:02 PM:	#9
Hi MJ Thanks for your response once again. "So, you won't find your answer within that domain of Algebra." Do you care to explain how this trifecta can be explained away within geometry(or elsewhere, perhaps metaphysics) since Algebra is impotent in this regard.

deusExMachinaV42
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Posted 02/10/09 - 04:48 PM:	#10
Turn to the direction you are not facing and walk backwards. Where are you going? Turn to the direction you are facing and walk forwards. Where are you going?

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