# Is math based on observations?

Axioms are chosen by common sense. Common sense is a product of what we observe in the environment we live in. So is math based on observations?

(Ok, this should have been under philosophy, but I can't find how to delete it...)

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EL said:
Axioms are chosen by common sense. Common sense is a product of what we observe in the environment we live in. So is math based on observations?
Only our application of math to the world is based on observation. The math itself is not.

When studying math, you can chose any axioms you want. How much those axioms describe reality is only relevant if you are trying to use math to describe reality. If you are simply studying the math itself then the actual "truth" of the axioms is irrelevant.

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Axioms are not chosen by what we observe in the environment we live in. Axioms are self-evident truths that cannot be proved within a system and are therefore given to be true.

For example, 2+2=4 is not based on our observation of objects or anything. If you take away the objects, does 2+2 still equal 4? Of course.

So to formally answer your question: no, math is not "based" on observations. We can use math to explain the things which we observe. For example, Newton used Calculus to explain the motion of objects. But he did not base his math on the time it took for a ball to fall from a five-story building.

master_coda said:
Only our application of math to the world is based on observation. The math itself is not.

When studying math, you can chose any axioms you want. How much those axioms describe reality is only relevant if you are trying to use math to describe reality. If you are simply studying the math itself then the actual "truth" of the axioms is irrelevant.
What about the games we play in math, called logic?
What is logically correct and what is wrong? How is that decided?

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philosophking said:
Axioms are not chosen by what we observe in the environment we live in. Axioms are self-evident truths that cannot be proved within a system and are therefore given to be true.
What decided what is self-evident or not?

For example, 2+2=4 is not based on our observation of objects or anything. If you take away the objects, does 2+2 still equal 4? Of course.
Doesn't that depend on what we define 2+2 to be? Isn't that a matter of taste?

matt grime
Homework Helper
If you have the stomach for it then read Russell and WHitehead to see it proven that 1+1=2.

We choose our axioms as the minimal set of rules we wish to work by. Situations decree that we may adopt different axioms, even the negation of some axioms, at different times to suit our needs. Just think of the parallel postulate of geometry, or the axiom of choice.

Logic is based on the rules of logic such as A=>B <=> (notA)OR(B) and the distributive property of disjunction and conjunction. (There are other systems that are studied; some people tihnk fuzzy logic may help in neural nets.) If you wish to use another system simlpy declare its rules, but be careful it's a very difficult thing to do.

matt grime said:
If you have the stomach for it then read Russell and WHitehead to see it proven that 1+1=2.
And there will go ten years of his life...

EL :This is a nice question !

When Witgenstein got his Ph.D for the book Taractatus ( which he wrote many year fefor during first world war) after the exam he told Russel who was one of the Professor there in this exam: : " don’t worry you will never understand it ..."

If you want a real answer to your question better that reading the prove 1+!=2 if you learn Witgenstein attitude to mathematics.

Moshek

EL said:
What about the games we play in math, called logic?
What is logically correct and what is wrong? How is that decided?
When we talk about things being logically correct we are talking about an argument being deductively valid. In other words, the conclusion is true if and only if the premises are true. Mathematics is a deductive system. But that's nothing interesting.

What is particularly important, as others have eluded to in this thread, is the premises we choose. Now, even if the premises are false, we can still have a deductively valid argument. For instance: Suppose the moon is made of green cheese and 1+1=2. Then it follows that the moon is made of green cheese or 1+1=2. That's a pretty trivial deductively valid argument. But it shows that a deductively valid argument does not have to be empirically true. This is what gives mathematics its power, it is independent of changing empirical knowledge.

In a sense, I think mathematics is circular. Strictly, it isn't, I can't emphasise that enough. But the axioms are defined because we would like certain consequences. In other words, we define our axioms because of what follows, and what follows is what it is because we have defined the axioms in such a way. In that sense, I think there is something to be said about mathematics being a natural science. Would we define our axioms in such a way so that they ruined Newtonian mechanics, Kepler's celestial mechanics, and so on?

moshek said:
EL :This is a nice question !

When Witgenstein got his Ph.D for the book Taractatus ( which he wrote many year fefor during first world war) after the exam he told Russel who was one of the Professor there in this exam: : " don’t worry you will never understand it ..."

If you want a real answer to your question better that reading the prove 1+!=2 if you learn Witgenstein attitude to mathematics.

Moshek
It's an interesting point bringing up the Tractatus. It does argue that propositions are isomorphic to states of affairs in the world, and that the constituent names of propositions designate objects in the world. So it would seem that mathematics is not independent of reality.

Of course, one of the goals of the Tractatus is to elucidate the link between language and reality. With a few seemingly plausible assumptions, we are led to conclude that the relationship between language and reality must be unconventional, that there must be some intrinsic nature of them that binds them together.

A few thousands of years ago, the first kind of mathematics developed was geometry, based on observations. After having a very good base, we can perform and develop new kinds of mathematics, and today is a sciencie which don't lies in the real world, but viceversa yes. That's the point. The world lies on mathematics, because several centuries ago, when the deductive model were born, mathematics began to left the real world, and today, maths are a world appart.

Bye

Stevo:

I bring it just for showing it is a waste of time to read Russel prove that 1+1=2.

You probably know that Wittgenstein himself understand that the Taractus is wrong . so he developed new attitude and we have 2 Wittgenstein.

Both of them were giant !

So he have the only one exact answer to EL question.

Moshek

moshek said:
Stevo:

I bring it just for showing it is a waste of time to read Russel prove that 1+1=2.

You probably know that Wittgenstein himself understand that the Taractus is wrong . so he developed new attitude and we have 2 Wittgenstein.

Both of them were giant !

So he have the only one exact answer to EL question.

Moshek
I certainly wouldn't wade through Russell's and Whitehead's Principia just to read that proof. I think there's better methods to get that sort of logical and philosophical lesson.

I think the criticisms Wittgenstein levelled against the Tractatus in the Investigations are quite strong, but I don't think they are conclusive. Of course, the assumptions which underly the Tractatus are similarly plausible, but I don't think they are necessarily true, a priori. I find there to be enticing arguments in both texts. Whether any theory which combines them is able to be consistent, I don't know.

EL : I like your question !

But let me ask you first which math do you ask if it base on observation ?

The regular and the linear way of thinking in mathematics do not base on observation but on analyzing the realty. But the realty is much more then the part it include this is way there is some general feeling that mathematics is very close to it dead end (proving theorems by computers etc)

The unity of Goedel theorem in some positive way can bring us to create a new mathematics, which will be base on observation and a deep understanding that we are part of the universe, and the universe is a part from us.

A very good reference to all this may be Goethe attitude to science. Well he did not appreciate so mach mathematics but he was thinking about the generic phenomena.

Very simple observation for example is that a page which we write mathematics on it is two dimension object which is more then 1.

In this new mathematics there is no place to binary logic the new center is the organic unity of mathematics which is a vision of Hilbert from 1900.

Best
Moshek

www.gurdjieff-internet.com/books_template.php?authID=121

Ok, maybe a bad choice of me to attack the axioms...I will go on with the logic instead. What I'm claiming is that math need not be something separable from how we observe the world, because math is based on logic, and our feeling for what is logically correct depends on how we see upon the world. I.e. logic is a human construction, based on our intuition for how it should be. And intuition is a product of how we experiences the world (i.e. what we observe...)

Matt Grime wrote "Logic is based on the rules of logic", which is in itself a contradiction. According to "logic" this is reasoning in a circle...

Matt continued: "If you wish to use another system simlpy declare its rules, but be careful it's a very difficult thing to do."
And how will we know how to follow these new rules? Isn't it true that we always in some way must fall back on our intuition of what is "correct" or not? (I don't mean that these new rules must be based on intuition, but our way to extract results from them must be...)

Stevo wrote: "Suppose the moon is made of green cheese and 1+1=2. Then it follows that the moon is made of green cheese or 1+1=2. That's a pretty trivial deductively valid argument."
How can we say that ´the moon is made of green cheese or 1+1=2´ follows from the first sentence? We use logic, i.e. our common sense, which is based on how we observe the world...

Note that I'm not in any way cracking down on math. Just questioning if it is based on our experiences or not?

What disturbes me a lot is that even my own reasoning here is paradoxial, since it's based on logic (or at least I tried...=))

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EL, it seems to me that what you are claiming is that our logical axioms are based on our intuition. I totally agree with you. However, I don't think it would be practical to have it any other way. Neither do I think that it is problematic for the validity of logic. The axioms of logic are independent of human judgement.

So please tell me ,which observation bring us to think that without logic there is no mathematics at all?

Integral
Staff Emeritus
Gold Member
I really have to simply say no to the original question. Math is based on axioms. There is no such thing as a point, a line or a plane these are all constructs of the human mind that only exist approximately.

Consider things like Matrix Algebra and Riemann's Geometry, both of which were developed with no apparent use, they waited patiently on the shelf until Physics and the physical world found uses for them, this is not observation this is human imagination.

Sure Diff equs. need physical boundary conditions if you want to model a physical situation but it is entirely possible to consider different classes of boundary conditions with out regard to the physical world. Entire classes of solutions can be explored in this manner with out ever computing a single number. That is the true power of Math.

matt grime
Homework Helper
There is nothing contradictory about saying logic is based on the rules of logic. All maths is based on its rules of practice. (Wittgenstein, for Moshek, a word is its use in language.)

Also you don't seem to appreciate that when you declare its rules you aren't saying its rules are true in any real sense. You may define another system where the rules are dramatically and contradictorily different. The way you 'extract' things from them is inherent in the rules of the system. (A implies B is equivalent to (notA) or (B) again.) Deductive reasoning is a natural (in some sense) process. It doesn't always produce th correct answers in real life but in maths it ought to.

Stevo said:
EL, it seems to me that what you are claiming is that our logical axioms are based on our intuition. I totally agree with you. /.../ The axioms of logic are independent of human judgement.
I don't understand your reasoning. Or is it just that you "believe" that the "axioms of logic" are independent of human judgement?

Integral: I don't think you got me right, and I can't blame you for that since my starting point with attacking the axioms was missleading. Sorry for that.
You wrote: "Math is based on axioms.", I mean that: Math is a game where we use logic to extract results out from axioms.
The axioms need not of course be fitted to the world around us. It's a choice of the creator of "the system".
However I claim that "logic" may be based on our intuition, which makes math a human construction, no mather how the axioms are chosen...

Matt grime wrote:"You may define another system where the rules are dramatically and contradictorily different. The way you 'extract' things from them is inherent in the rules of the system. (A implies B is equivalent to (notA) or (B) again.) Deductive reasoning is a natural (in some sense) process".
(I Think) I get your point but how do you know how to follow the rules of this new logic system? Surely you need to know what for example "imply" and "equaivalent" means to be able to follow the schedule. At some point we have to fall back on our intuition. (Now you may say that what is ment by these word is just a definition, but how do we then know what a definition is, and how we can use it?)

moshek wrote: "So please tell me ,which observation bring us to think that without logic there is no mathematics at all?
I'm not sure I get you?
Are axioms without any way to extract results from them mathematics?

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matt grime
Homework Helper
Two logical constructs are defined to be equivalent if they have the same truth table. You know how to follow these rules because you are told how to follow them. As to why you follow them, well that is a different matter. THere are many schools of logic, the one that mathematics tends to adhere to is one which adopts the excluded middle. At no point do we need to use our intuition. A mathematical object is to all intents and purposes its defintion, which is its defining properties. You have no safely gone away from mathematics into philosophy, and as such no mathematical answer will really suffice, because there might not be one. In fact how do we even know you're asking the correct question?

matt grime said:
Two logical constructs are defined to be equivalent if they have the same truth table.
So when are they "the same"?

You know how to follow these rules because you are told how to follow them.
And how do I know how to follow the instructions of how to follow these rules?

You have no safely gone away from mathematics into philosophy, and as such no mathematical answer will really suffice, because there might not be one. In fact how do we even know you're asking the correct question?
As I wrote from the beginning, I agree that this thread should be under "philosophy".

matt grime
Homework Helper
Now it appears you are just asking questions for the sake of asking, questions like a child who's learned the word why. If you can't tell when two truth tables are the same, or how to follow deductive reasoning then there's little hope of finding a solution to this is there?

Matt:

Didn't Einstein ask question of a child so he could develop relativity ?
If your answer is yes, then way you blame EL for trying to understand something deeply about mathematics. In regular mathematics even with axioms and logic there is always some point that is taken for granted . mathematician name it presumptions and I call it the blind point of any Euclidian mathematics. Please answer me for that and then i will come to talk with you about Wittgenstein and mathematics.

Moshek