Mathematics, Truth, Consensus And Opinion

[TH300]

I am referring to a post by Hidiot:

Also, two plus two equals four - don't try to tell me that this is a statement of opinion and it has no truth value.

Although you will probably laugh: Yes, it is a statement of opinion. Mathematics are made up by humans and are thus subject to opinion as is anything else which is made by human. If you believe that you can prove me wrong, you should do so in the debate forum.

Math only works because there is a (global) consensus on how it works. And this consensus is strengthened by the fact that math can be used to model our perceived reality.

Also, base 10 was chosen by consensus to be the base in which most elementary-ish math is done.

So, math is basically a consensus, NOT an opinion.

The basic claim in this post is that consensus is not opinion. This is wrong, because consensus is just another word for general agreement, i.e. everyone has the same opinion or does at least not speak up against it. Consensus can change and can differ in different cultures. One prominent example is the axiom of choice which isn't acknowledged by all people who work with mathematics. It is proven that mathematics are free of contradictions with or without this axiom, if one assumes that the other axioms are not contradictory. Hence its really only subject to opinion. We don't know whether mathematics are free of contradictions.

Even if you claim that mathematics must be right, because they describe physics correctly, you cannot prove this claim, because you don't know all about physics and there is a chance that one day some aspect about physics is found that doesn't fit together with our existing mathematics.

[Freeza-CII]

Even if you claim that mathematics must be right, because they describe physics correctly, you cannot prove this claim, because you don't know all about physics and there is a chance that one day some aspect about physics is found that doesn't fit together with our existing mathematics.

Im not trying to be rude. But by this quote you yourself would have to know all there is to know about the known universe to know this is accurate.

Personally i feel mathmatics is pretty solid it explains things to the way we see them now. a matter of perception perhaps. But with the more complicated often comes with more changes. But alot of things are agreed on in a group in the realm of science. This is how we get the laws such at the boils and pascal ect. and theories such at the theory of relitivity.  

[jcj94]

1: Agreeing with BOTH of you.

Black Holes and galaxies both defy some laws, Just like the very very tiny.

But, yes, mathamatics is solid.  Depening on what base you use (2, 10,
2+2=4 Dec
10+10=4 Bin  
2+8=A Hex

Its still solid, following the same pattern of logic.  

[TH300]

Even if you claim that mathematics must be right, because they describe physics correctly, you cannot prove this claim, because you don't know all about physics and there is a chance that one day some aspect about physics is found that doesn't fit together with our existing mathematics.

Im not trying to be rude. But by this quote you yourself would have to know all there is to know about the known universe to know this is accurate.

In this case I am not talking about the real chance of finding such inconsistency, but about the perceived chance. I.e. from our perspective, we cannot prove that current mathematics describe all physical laws (even those we don't know) correctly. Since we cannot prove that, it can either be true or not be true, i.e. the perceived probability for each option is not zero.

Personally i feel mathmatics is pretty solid it explains things to the way we see them now. a matter of perception perhaps. But with the more complicated often comes with more changes. But alot of things are agreed on in a group in the realm of science. This is how we get the laws such at the boils and pascal ect. and theories such at the theory of relitivity.

The fact (I'm not even sure, if it is a fact) that mathematics work flawlessly until the current date, doesn't mean, they'll continue to do so. Thats like saying "I have seen only white sheep until now, hence there are only white sheep"

JCJ, mathematics is more than 1+1=2, a lot more.

[CK9]

Why do we call facts facts?  We all observe them to be proven true time and time again.

A consensus is an agreement to believe in the same thing (according to the mirriam-webster dictionary).  So, looking at that as a basis, we can conclude that our choice in numeral-base system is an opinion.  However, this does not prove math to be an opinion.

In decimal, we count as 1, 2, 3, 4, 5...ad infinitum

These numerals (both written and oral) are merely a representation of a concept.  We count objects in this manner, and will always get the same result regardless of the bas-system chosen (that is to say, just because the chosen representation sounds and/or looks differnt, it doesn't change the actual ammount of the objects we are counting).

So, keeping that point in mind, we can now say math is, indeed, fact and not opinion.  In decimal, when we have 2 of an object added with 2 more of an object, we will always end up with 4 of that object.  In binary, we represent this as 10 + 10 = 100.  The representation is different, but the actual action is the same.

If we use the basis of math as an opinion, then I could say 1 + 50 = 5 in decimal.  This is impossible, as the physical objects being represented by these figures do not work in this manner.

And that's only going over the most basic mathematical systems.  We have calculus, triganometry, laplace transforms (my personal least favorite -.-), matrix math...there are so many equations all because they have been shown to always* be true.

(* always is to be taken loosly, as there are cases where an equation is wrong for a set of examples, which leads to changes to the equation to explain the variance in results (such as in the many principles, theorems, and laws I had to learn to earn my degree, heh))

[jcj94]

And to go along with the expection that CK9 was talking about:

What do you think the number i and e are?

They are both unreal, yet common occurences in math, just lik pi being an irrational number.  sqrt(-1)=i and I can't remember the equation to get e off the top of my head.

They are irrational, but have rules based around them because they make sense (you can't have a square root of a negative number because - times a - always equals a positive, and sometimes, when a parabola doesn't go across the x axis, you will get a negative number under the sqrt sign. Thus, the # i^2 is used and multiplied underneath to get a number you can divide further and make the equation level out.)

[Hooman]

Math is a theoretical construct. It can be perfectly true and self consistent in it's own right, whether or not it describes the real world, or has any physical realizations. It just wouldn't be nearly as useful if it didn't.

Also, if you don't make some assumptions about the world, you can't draw any useful conclusions about it. Sure, you can doubt math, or your ability to see or perceive the world about you, or you can doubt your own existance, but doing so doesn't appear to be very useful. Indeed, if you refute a claim such as "I think, therefor I am", what action is left to do? Or if you deny logic, what point is there in trying to have a logical conversation about logic? It would seem illogical to try. But then, if you're refuting logic, I suppose that doesn't really matter.



Edit: As for that imaginary number bit. What exactly is imaginary about an imaginary number? Is it perhaps just an unfortunate name? Certain that system of math describes a number of observed physical phenomenon quite well.

Also, what's "real" about e and Pi? Consider this: You take a measurement of the "real" world. That measurement has finite precision. That makes it a rational number. There is no way to measure the real world and get a "real" number (that isn't strictly rational). Are there really such things as irrational numbers? Perhaps irrational numbers are a made up mathematical concept that have no physical realization, and simply don't appear to contradict the real world in any obvious way. Perhaps the rational imaginary numbers, as I shall call them, are more "real" than real numbers.

Btw, I've heard of some work where Calculus was being worked out in a system in which real numbers didn't exist. It basically sounded like you only needed the rational numbers to do Calculus. But then, I've also seen "constructions" of the other numbers sets, given the positive integers as a starting point. I suppose you could just construct the "real" numbers if you needed them, but then, it didn't sound like you needed them.