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#1 Skeptic

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Posted 20 June 2005 - 03:16 AM

Question: can mathematics be labeled "objectively true"?

I made these two very important statements:

1) Since mathematics is a non-empirical system of symbolic logic, it relies on a system of (logical and non-logical) axioms.

2) At the level of axioms, no system of logic can be said to be "true".

Just to clarify exactly what I am saying about the truth of mathematics: at the level of axioms, mathematics cannot be said to be true.

Simple as that.

If any part of that assertion is not clear, please show which part and I will try my best to explain it. :bye:

#2 Amy Parkin

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Posted 20 June 2005 - 03:35 AM

But all these axioms can be proved to be true. :confused:

#3 Flappie

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Posted 20 June 2005 - 03:41 AM

But all these axioms can be proved to be true. :confused:

Not really, you can agree they are true though.
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#4 Amy Parkin

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Posted 20 June 2005 - 03:45 AM

But all these axioms can be proved to be true.  :confused:

Not really, you can agree they are true though.

But how did people work them out in the first place? They didn't just make them up.

#5 DJP

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Posted 20 June 2005 - 04:01 AM

I agree with Amy.

I am yet to see an axiom that can't be proved to be true.

DJP

#6 tarkus

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Posted 20 June 2005 - 04:36 AM

Skeptic's point looks like Godel in a soundbite. Or is there more behind this?

T

#7 DJP

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Posted 20 June 2005 - 04:59 AM

I can't see a connection with Gödel.

And I reckon this:

1) Since mathematics is a non-empirical system of symbolic logic...

is empirically false. :) Maybe I'm being naughty.

DJP

#8 Skeptic

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Posted 20 June 2005 - 05:04 AM

Guys

Here is an explanation of the difference between a logical and a non-logical axiom, both of which are forms of axioms used in mathematics.

The explanation starts off by referring to the way an axiom is usually understood in epistemology, as well as the controversy inherent in that understanding:

In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. To say the least, not all epistemologists agree that any axioms, understood in that sense, exist.


The point is then made that:

In mathematics, axioms are not self-evident truths. They are of two different kinds: logical axioms and non-logical axioms . Axiomatic reasoning is today most widely used in mathematics.


The explanation on logical axioms makes the point that these types of axioms used in mathematics, come closest to being "true", but (take note, DJP) that the mathematical requirement of offering "proof" of an axiom, contradicts the very notion of the "self-evident truth" of an axiom:

These are formulas which are valid, i.e., formulas that are satisfied by every model (a.k.a. structure) under every variable assignment function. More colloquially, these are statements that are true in any possible universe, under any possible interpretation and with any assignment of values.

Now, in order to claim that something is a logical axiom, we must know that it is indeed valid. That is, it might be necessary to offer a proof of its validity (truth) in every model. This might challenge the very classical notion of axiom; this is at least one of the reasons why axioms are not regarded as obviously true or self-evident statements.

Logical axioms, being mere formulas, are devoid of any meaning; but the point is that when they are interpreted in any universe, they will always hold no matter what values are assigned to the variables. Thus, this notion of axiom is perhaps the closest to the intended meaning of the word: that axioms are true, no matter what.


But it is when we look at non-logical axioms, that it becomes really clear that such axioms can in no way be "true":

Non-logical axioms are formulas that play the role of theory-specific assumptions. Reasoning about two different structures, for example the natural numbers and the integers, may involve the same logical axioms; the non-logical axioms aim to capture what is special about a particular structure (or set of structures, such as algebraic groups). Thus non-logical axioms, unlike logical axioms, are not tautologies. Another name for a non-logical axiom is postulate.

Almost every modern mathematical theory starts from a given set of non-logical axioms, and it was thought that in principle every theory could be axiomatized in this way and fomalized down to the bare language of logical formulas. This turns out to be impossible and proved to be quite a story.

This is the role of non-logical axioms, they simply constitute a starting point in a logical system. Since they are so fundamental in the development of a theory, it is often the case that they are simply referred to as axioms in the mathematical discourse, but again, not in the sense that they are true propositions nor as if they were assumptions claimed to be true. For example, in some groups, the operation of multiplication is commutative; in others it is not.


Note that it says "almost every modern mathematical theory starts from a given set of non-logical axioms" and from the discussion above, we know that non-logical axioms are theory-specific. In other words, unlike logical axioms, non-logical axioms are not valid in all contexts. Clearly, something that is not true "in any possible universe, under any possible interpretation and with any assignment of values" cannot be claimed to be objectively "true".

Hopefully it is becoming clearer now why I am saying that at the level of axioms, mathematics cannot be said to be true.

Edited by Skeptic, 20 June 2005 - 05:09 AM.


#9 Skeptic

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Posted 20 June 2005 - 05:21 AM

DJP

And I reckon this:

1) Since mathematics is a non-empirical system of symbolic logic...

is empirically false. :) Maybe I'm being naughty.


Very naughty. And for that, you'll have to pay...! <manic laughter>

BTW, are you saying...

1) Mathematics is empirical
2) mathematics is not a system of symbolic logic
3) Both

:?:

#10 Flappie

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Posted 20 June 2005 - 05:25 AM

Could you give me an understandable example?
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#11 Skeptic

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Posted 20 June 2005 - 06:03 AM

Now, the moment someone claims that mathematics cannot be called "true", someone else is bound to challenge that person to say whether 1+1=2...

To answer that question, one needs to ask whether 1+1=2 is an axiom or not.

Stated in that form, it is not an axiom (although it could be). So saying 1+1=2 is completely valid. But is that the end of it? Unfortunately not...

When we refer to 1+1=2, there are several axioms which we are assumed to understand implicitly, but which we rarely think of explicitly...

For instance:

  • There is a natural number 0.


  • Every natural number a has a successor, denoted by a + 1.


  • There is no natural number whose successor is 0.


  • Distinct natural numbers have distinct successors: if a <> b, then a + 1 <> b + 1.


  • If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers.


If you think that is the end of it, then check out the difference between natural numbers, integers, rational numbers, real numbers, complex numbers, algebraic numbers and transcendental numbers here.

So, we have established that it is valid to say 1+1=2, right.

Now, what if I were to ask you to say whether this is valid: 1+1=10....?

Of course not! Not if I am using the decimal* system anyway!

But if I was using the binary system it would be completely valid to say 1+1=10...!

Enough said?

*The decimal ystem happens to be based on the number 10 (deca means ten), which is simply because human beings counted on their ten fingers long before they could write. There is no special need for the system to be based on ten, as the existence of other systems, like binary, hexadeximal, etc, show...

Edited by Skeptic, 20 June 2005 - 06:12 AM.


#12 DJP

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Posted 20 June 2005 - 06:17 AM

Note that it says "almost every modern mathematical theory starts from a given set of non-logical axioms" and from the discussion above, we know that non-logical axioms are theory-specific. In other words, unlike logical axioms, non-logical axioms are not valid in all contexts. Clearly, something that is not true "in any possible universe, under any possible interpretation and with any assignment of values" cannot be claimed to be objectively "true".

Multiplication is a good example. Multiplication is commutative if we speak of integers, but it is not commutative if we speak of matrices.

"Multiplication of integers is commutative" is empirically true. I.e. we empirically find out that there are such things as integers, and we empirically find out that there is merit in combining them in the way we customarily call "multiplying." This is not at all obvious, but we learn it young. It took me a long time to accept that multiplying things was interesting and useful, I vaguely remember the struggle.

"Multiplication of matrices is not commutative" is empirically true. We empirically find out that there are such things as matrices, and we try to treat them with our familiar tools. We look for something like "multiplication" that we know and love, but find that it's taken a different guise. We only call it "multiplication" because of a valuable analogy with integers. Noncommutative multiplication is empirically true because the real world really fits it. Ditto commutative multiplication.

At no stage do we have to decide whether the statement "multiplication is commutative" is objectively true or false. The question doesn't arise: it's too poorly specified to be useful. The idea of "context dependence" is not a good one: it isn't a matter of "true here, not true there => conflict", but simply that we humans oversimplify things by trying to call a diverse set of operations "multiplication."

You can come up with whatever weird axioms you like, and I reckon you'll find that it describes an aspect of the real world that is valuable to somebody somewhere. What I mean is this: I seriously doubt whether there is any mathematical theory at all that can't be considered a study of the real world.

And that could well be a physicist's bias and ignorance.

DJP

#13 Skeptic

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Posted 20 June 2005 - 06:26 AM

Flappie

Could you give me an understandable example?


When you see the digits "10" written on a page, do you pronounce that as "ten" or "two"?

I assume you answered "ten"...

In order to answer "ten" you have assumed that the decimal system applied. If you were a software engineer, you could have read the "10" to mean "two" and that would have been equally valid, considering that the person was working in the binary system...

So here we have an interesting phenomenon: the digits "10" could mean either "ten" or "two".

Equally, depending on the system used, it could be equally valid to say "1+1=10" as it is to say "1+1=2".

Now, Flappie, keeping that in mind, is it reasonable to say "1+1=10" is a universally true statement?

Edited by Skeptic, 20 June 2005 - 06:50 AM.


#14 Flappie

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Posted 20 June 2005 - 07:00 AM

Isn't that just a matter of sticking a (base 2) at the end?
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#15 Flappie

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Posted 20 June 2005 - 07:03 AM

I often wonder how the Romans managed to multiply, but that's a bit offtopic.
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#16 Skeptic

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Posted 20 June 2005 - 07:30 AM

DJP

"Multiplication of integers is commutative" is empirically true.  I.e. we empirically find out that there are such things as integers, and we empirically find out that there is merit in combining them in the way we customarily call "multiplying."  This is not at all obvious, but we learn it young.  It took me a long time to accept that multiplying things was interesting and useful, I vaguely remember the struggle.

"Multiplication of matrices is not commutative" is empirically true.  We empirically find out that there are such things as matrices, and we try to treat them with our familiar tools.  We look for something like "multiplication" that we know and love, but find that it's taken a different guise.  We only call it "multiplication" because of a valuable analogy with integers.  Noncommutative multiplication is [color=red]empirically true because the real world really fits it. [/url] Ditto commutative multiplication.

*snip*

And that could well be a physicist's bias and ignorance.


In order to talk about discovering anything in a truly empirical fashion, there needs to be something physical to discover, right? Remember, the empirical method implies discovering something that exists, not on some abstract level, but on a physical level. That is why, for instance, philosophy and large parts of psychology are non-empirical.

Mathematics is non-empirical to the extent that in generating a body of knowledge of mathematics, one in no way has to rely on empirical observation at all. Empirical observation is at most a peripheral, non-essential part of mathematics.

Here's the essence:

The major problem of mathematical realism is this: precisely where and how do the mathematical entities exist? Is there a world, completely separate from our physical one, which is occupied by the mathematical entities? How can we gain access to this separate world and discover truths about the entities? Gödel's and Plato's answers to each of these questions are much criticised.


To refer to your example of multiplication: yes, you discovered the rules of multiplication, but those rules aren't physical entities. They are abstract. So what you term your "empirical" discovery of these rules, simply refers to your discovery that these rules "work" or "apply".

Sigmund Freud's notion of the "Id, Ego and Superego" fits the facts perfectly, but they have never been empirically verified and the simple fact is that they can't be.

#17 Skeptic

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Posted 20 June 2005 - 07:34 AM

Flappie

I often wonder how the Romans managed to multiply, but that's a bit offtopic.


Not at all off topic. The Roman numerical system is wholly unsuitable for multiplication. That is why our (Western) ancestors kept their alphabet, but adopted the system of Arabian numerals. Take a close look and you can actually see that 1, 2, 3, 4, 5, etc looks Arabic.... :shades:

Now there is a conspiracy for you!

#18 Skeptic

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Posted 20 June 2005 - 07:37 AM

Flappie

Isn't that just a matter of sticking a (base 2) at the end?


Correct. And if you have to stick a "base 2" at the end in order for the statement (1+1=10) to be valid, it can't be a universally true statement, can it?

#19 mji2

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Posted 20 June 2005 - 10:49 AM

I often wonder how the Romans managed to multiply, but that's a bit offtopic.

Your mind could do with filling a bit more Flappie.

#20 Flappie

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Posted 20 June 2005 - 10:57 AM

No, what I meant was though is that 1+1=10base2 (or whatever the proper notation is) would be a universally true statement, just a matter of shrinking the universe for it to be true. :P


I was aware of the Arab numeral thing, had to write an Arab-Roman numeral (and vice versa) convertor a while ago.
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