Math is a language, not a science?

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Re: Math is a language, not a science?

Postby eSOANEM » Mon Jun 06, 2011 7:56 pm UTC

Find me a definition of "proven" from a reputable source that doesn't require truth. If "P" and "not P" are both shown to follow from the axioms of a system, neither can be true (as it being so would contradict the truth-iness of the other one which has just as much right to being true) and so neither can be proven and to say otherwise is to rob the word "proven" of any useful meaning.
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Re: Math is a language, not a science?

Postby skeptical scientist » Mon Jun 06, 2011 10:00 pm UTC

eSOANEM wrote:If "P" and "not P" are both shown to follow from the axioms of a system, neither can be true (as it being so would contradict the truth-iness of the other one which has just as much right to being true) and so neither can be proven and to say otherwise is to rob the word "proven" of any useful meaning.

This is true. Hence, the fact that P is proven in inconsistent system Y is meaningless, since that fact is equally true of every other proposition. However, the proof/derivation itself has meaning, since it tells you exactly how the inconsistent axioms entail proposition P. One possible application of this meaning is when the same proof can be carried out in another system Z, which may be consistent.

One way to make this distinction precise is the following observation: if I tell you that system Y is inconsistent, you can immediately conclude that P is provable in system Y. So once you know that Y is inconsistent, saying that P is provable provides no new information. You cannot immediately produce a proof of P in system Y, however. So a proof of P in Y is new information.
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Re: Math is a language, not a science?

Postby mike-l » Mon Jun 06, 2011 10:22 pm UTC

nachomancer wrote:Imagine, for instance, that I have shown that a certain class of axiomatic systems (perhaps including, say peano arithmatic) contains only inconsistent systems, but that the shortest contradiction has length (according to the metric given by number of applications of an axiom or something) far larger than the size of the universe. We could still make meaningful statements as long as they are shorter than the shortest contradiction!


Not really. Suppose you have a consistent system X which proves P, but the shortest proof is k>1 symbols/applications/etc long. Then X + not P is an inconsistent system with no contradiction with k or fewer symbols/applications/etc. not P is a 1 step proof, but surely not meaningful. A proof of P using k+1 symbols and not using (not P) though would be meaningful (Here I'm agreeing with the notion that if a proof works with a subset or modification of the axioms that are themselves consistent then it's meaningful)
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Re: Math is a language, not a science?

Postby Token » Mon Jun 06, 2011 10:39 pm UTC

eSOANEM wrote:Find me a definition of "proven" from a reputable source that doesn't require truth. If "P" and "not P" are both shown to follow from the axioms of a system, neither can be true (as it being so would contradict the truth-iness of the other one which has just as much right to being true) and so neither can be proven and to say otherwise is to rob the word "proven" of any useful meaning.

You've got things backwards. The standard definition of "proven" (in the sense of mathematical logic) says nothing about truth - it means that something has been derived from a particular collection of axioms via a particular collection of inference rules. It's just a property that results from rearranging strings of symbols in a particular, ordered way.

The reason truth comes into it is because we attach meaning to these symbols that we've manipulated - they represent not just markings on a page, but abstract statements about conceptual structures. It is these statements that can be true or false, not the strings of symbols, but we are able to "pull back" their truth or falsehood along our interpretation, and attach it to the symbols themselves - and consequently also attach some relevance to the process of proving itself.

So when we have proved both "P" and "not P" in the context of a formal system, it's not that neither is true - but that the inconsistency of the system means that our interpretation was necessarily wrong. And since we have no interpretation, P becomes a contextless string of symbols again, to which no truth value can be meaningfully attached.

skeptical scientist wrote:One way to make this distinction precise is the following observation: if I tell you that system Y is inconsistent, you can immediately conclude that P is provable in system Y. So once you know that Y is inconsistent, saying that P is provable provides no new information. You cannot immediately produce a proof of P in system Y, however. So a proof of P in Y is new information.

The problem being, of course, that the usual proof of P in an inconsistent system Y carries no more information than the inconsistency of Y. Only if the proof of P does not rely on the inconsistency of Y do we get any new information, but for that information to be really useful it would have to come from some subset of consistency in Y. So I think what we can conclude from the last page-and-a-half is that "inconsistent systems can be useful, but only insofar as they can be restricted to subsets of their axioms which *are* consistent".
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Re: Math is a language, not a science?

Postby skeptical scientist » Mon Jun 06, 2011 11:57 pm UTC

Token wrote:
skeptical scientist wrote:One way to make this distinction precise is the following observation: if I tell you that system Y is inconsistent, you can immediately conclude that P is provable in system Y. So once you know that Y is inconsistent, saying that P is provable provides no new information. You cannot immediately produce a proof of P in system Y, however. So a proof of P in Y is new information.

The problem being, of course, that the usual proof of P in an inconsistent system Y carries no more information than the inconsistency of Y.

Of course it does, because a proof of a contradiction carries more information than the mere fact of inconsistency. It's only the last two lines of a proof of P which are predictable, and thus give no new information (and then only if the proof of P goes through contradiction). This information can be useful. For example, if you're trying to fix the inconsistent system, knowing how you prove a contradiction helps you go about this (say, if Bertrand Russel just wrote you a letter saying that your basic law v leads to paradox).
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Re: Math is a language, not a science?

Postby Token » Tue Jun 07, 2011 9:40 am UTC

If Bertrand Russell bothers to tell you what the paradox is, proving P from it tells you nothing useful. Unless you're saying it's possible to deduce the inconsistency of Y without also having enough information to derive a contradiction from its axioms?
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Re: Math is a language, not a science?

Postby gmalivuk » Tue Jun 07, 2011 3:48 pm UTC

Token wrote:If Bertrand Russell bothers to tell you what the paradox is, proving P from it tells you nothing useful.
Well sure, because in telling you "what the paradox is", Russell already did that work for you.

But if you can't find out what the paradox is without actually running the proof that leads to it, then of course running that proof tells you something useful.
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Re: Math is a language, not a science?

Postby Token » Tue Jun 07, 2011 4:22 pm UTC

gmalivuk wrote:But if you can't find out what the paradox is without actually running the proof that leads to it, then of course running that proof tells you something useful.

My point being, if you don't know what the paradox is or how to derive it, how do you know that there's a paradox? Or, to rephrase, if you don't have a proof of a paradox (or a known means of constructing one), how do you know that the system is inconsistent?
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Re: Math is a language, not a science?

Postby skeptical scientist » Tue Jun 07, 2011 4:38 pm UTC

Does it matter? You were told by a reliable source. You designed a quantum computer that searches through every derivation simultaneously and stops when it reaches a contradiction, but due to hardware limitations is unable to tell you what the problematic derivation was. Whatever. In any case, if you only know the system is inconsistent, without knowing a proof of a contradiction, then learning a proof provides new information.

At that point, a proof of P within the system only has meaning insofar as it can be carried out in some consistent subsystem, as you say, because once you know a proof of a contradiction, you can immediately produce a proof of P that goes through contradiction.
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Re: Math is a language, not a science?

Postby Token » Tue Jun 07, 2011 9:17 pm UTC

skeptical scientist wrote:Does it matter? You were told by a reliable source. You designed a quantum computer that searches through every derivation simultaneously and stops when it reaches a contradiction, but due to hardware limitations is unable to tell you what the problematic derivation was. Whatever.

Sigh. Why not just say "yes" to the question "Unless you're saying it's possible to deduce the inconsistency of Y without also having enough information to derive a contradiction from its axioms?"? Then it's clear we're using the word "know" differently - I mean "have a proof of", and you mean "have reason to suspect, through probabilistic means or appeals to authority" - and we can agree that this argument is pointless and go do better things with our time.
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Re: Math is a language, not a science?

Postby Talith » Tue Jun 07, 2011 11:35 pm UTC

You have proved the existence of an inconsistent statement though non-constructive means. Why do you seem to think knowing something exists means we can explicitly construct it? There are hundreds of existence proofs that don't explicitly construct the object you show to exist.

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Re: Math is a language, not a science?

Postby Token » Wed Jun 08, 2011 2:38 pm UTC

Talith wrote:You have proved the existence of an inconsistent statement though non-constructive means. Why do you seem to think knowing something exists means we can explicitly construct it? There are hundreds of existence proofs that don't explicitly construct the object you show to exist.

But there is a conceptual gap between proofs of statements of the form [imath](\exists x) \phi(x)[/imath], and (meta-)proofs that proofs of statements of a particular form exist. There are techniques for the former that just don't apply to the latter. Now, I haven't actually got a (metameta-)proof that you can't have a non-constructive (meta-)proof, but I'd be quite surprised if you could.
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Re: Math is a language, not a science?

Postby skeptical scientist » Wed Jun 08, 2011 4:44 pm UTC

Token wrote:But there is a conceptual gap between proofs of statements of the form [imath](\exists x) \phi(x)[/imath], and (meta-)proofs that proofs of statements of a particular form exist. There are techniques for the former that just don't apply to the latter. Now, I haven't actually got a (metameta-)proof that you can't have a non-constructive (meta-)proof, but I'd be quite surprised if you could.

Examples do exist, and can take the form where a stronger language/theory is proved to be conservative over a weaker language/theory, i.e. any sentence in the weaker language which is provable in the stronger theory is provable in the weaker theory. (An example would be the conservativity of ACA0 over PA. This is easily proved using the completeness theorem, since any model of PA can be turned into a model of ACA0 that makes the same first-order statements true by adding a second-order part consisting of the definable subsets of the model.) This statement plus any proof in the stronger theory constitutes a proof that there exists a proof in the weaker theory, but there may not be a good way of producing a proof in the weaker theory from this metaproof.

I asked this question on mathoverflow, and one of the examples I received was:
Emil Jeřábek wrote:One example which springs to mind is Jacobson’s theorem: if R is a ring such that for every aR there exists an integer n>1 such that a=an, then R is commutative. By completeness of equational logic, this implies that for any n>1, there exists an equational derivation of xy=yx from the axioms of rings and xn=x. Already finding such derivation for n=3 is a nontrivial exercise; explicit derivations are known for some n, but not in general.


Edit: while on mathoverflow, I also noticed this question, which is relevant to our discussion.
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Re: Math is a language, not a science?

Postby nachomancer » Wed Jun 08, 2011 8:57 pm UTC

skeptical scientist wrote:Edit: while on mathoverflow, I also noticed this question, which is relevant to our discussion.


I was just about to post this! I was happy to see that people with far more knowledge of logic than me can confirm that my thinking was sound. Here is what Gowers had to say (blatant argument by authority):

" if a theory has only very long contradictions, then I would have thought it might well have a structure that is in some sense "locally" a model (a bit like the surface of the Earth being locally a model for an infinite plane). "

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Re: Math is a language, not a science?

Postby skeptical scientist » Wed Jun 08, 2011 9:02 pm UTC

I was just about to post this! I was happy to see that people with far more knowledge of logic than me can confirm that my thinking was sound. Here is what Gowers had to say (blatant argument by authority):

" if a theory has only very long contradictions, then I would have thought it might well have a structure that is in some sense "locally" a model (a bit like the surface of the Earth being locally a model for an infinite plane). "

It's hard to debate the credentials of Gowers as an authority. That said, his statement contains many weasel-words, and (as far as I know, and as suggested in that thread) nobody has found a way to make his statement precise.
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Re: Math is a language, not a science?

Postby WarDaft » Wed Jun 08, 2011 9:16 pm UTC

Can we construct a system with the express purpose of it being inconsistent and having a contradiction derived from a proof of length N but no less, and then see how proofs of less than N look with as far as 'actual' truth is concerned? Or is it simply too hard to aim with regards to the size of N?
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Re: Math is a language, not a science?

Postby mike-l » Wed Jun 08, 2011 9:50 pm UTC

WarDaft wrote:Can we construct a system with the express purpose of it being inconsistent and having a contradiction derived from a proof of length N but no less, and then see how proofs of less than N look with as far as 'actual' truth is concerned? Or is it simply too hard to aim with regards to the size of N?


I already did this

mike-l wrote:
nachomancer wrote:Imagine, for instance, that I have shown that a certain class of axiomatic systems (perhaps including, say peano arithmatic) contains only inconsistent systems, but that the shortest contradiction has length (according to the metric given by number of applications of an axiom or something) far larger than the size of the universe. We could still make meaningful statements as long as they are shorter than the shortest contradiction!


Not really. Suppose you have a consistent system X which proves P, but the shortest proof is k>1 symbols/applications/etc long. Then X + not P is an inconsistent system with no contradiction with O(k) or fewer symbols/applications/etc. not P is a 1 step proof, but surely not meaningful. A proof of P using k+1 symbols and not using (not P) though would be meaningful (Here I'm agreeing with the notion that if a proof works with a subset or modification of the axioms that are themselves consistent then it's meaningful)


(Added O(k) to take care of things like a proof by contradiction of P becoming a shorter contradiction after adding not P to the axioms)
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Re: Math is a language, not a science?

Postby skeptical scientist » Wed Jun 08, 2011 10:34 pm UTC

mike-l wrote:(Added O(k) to take care of things like a proof by contradiction of P becoming a shorter contradiction after adding not P to the axioms)

Since (¬P -> P) -> P is a tautology, your O(k) can be k±O(1), where O(1) is probably something like 2 (depending on the specifics of your deduction system).
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Re: Math is a language, not a science?

Postby eSOANEM » Thu Jun 09, 2011 5:21 pm UTC

skeptical scientist wrote:
eSOANEM wrote:If "P" and "not P" are both shown to follow from the axioms of a system, neither can be true (as it being so would contradict the truth-iness of the other one which has just as much right to being true) and so neither can be proven and to say otherwise is to rob the word "proven" of any useful meaning.

This is true. Hence, the fact that P is proven in inconsistent system Y is meaningless, since that fact is equally true of every other proposition. However, the proof/derivation itself has meaning, since it tells you exactly how the inconsistent axioms entail proposition P. One possible application of this meaning is when the same proof can be carried out in another system Z, which may be consistent.


But then it only has meaning because of that other system. If there was no system in which P could be proven it wouldn't be true in any meaningful sense so any value attached to P is dependent on a consistent system in which P has that value.

Token wrote:
eSOANEM wrote:Find me a definition of "proven" from a reputable source that doesn't require truth. If "P" and "not P" are both shown to follow from the axioms of a system, neither can be true (as it being so would contradict the truth-iness of the other one which has just as much right to being true) and so neither can be proven and to say otherwise is to rob the word "proven" of any useful meaning.

You've got things backwards. The standard definition of "proven" (in the sense of mathematical logic) says nothing about truth - it means that something has been derived from a particular collection of axioms via a particular collection of inference rules. It's just a property that results from rearranging strings of symbols in a particular, ordered way.


If P follows from your axioms (which are assumed to be true) then P is true, "proven" has to imply truth if the axioms are true which is assumed when you define the system.
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Re: Math is a language, not a science?

Postby Dr beer » Fri Jul 12, 2013 8:43 pm UTC

A) Does anyone still monitor this thread?
B) I am not a mathmatician as will soon become evident

I am seeking help with what is essentially a language problem that utilises arithmatic symbols but appears to have a mathmatical/arithmatical consequence.

If for example I felt some basic arithmatic symbols represented more than one function and I were to seperate these functions and give them their own symbol under the basic premis (axiom) that all symbols and integers used must posses what linguistically is known as an extra linguistic capacity,( exists physically etc ) then it would follow that the results of sums under this axiom to hold truth must also be physically possible ( now I am talking extreme simplicity here, eg a man has two apples in each hand clearly has four apples (2+2+4) , real as simple as it gets stuff). Therefore if I was to run some fairly simple sums that result in an irrational (for example sq root of 2) we could conclude that the result lacked truth if had no physical aspect , but if as a consequence of this seperation of function the same sum generated a rational (whole number) result, clearly as a result of its extra linguistic capacity it would have truth and therefore hold precedence.

Now I am not saying this has been done but in principle is there a flaw in this logic?

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Re: Math is a language, not a science?

Postby Cleverbeans » Fri Jul 12, 2013 11:27 pm UTC

Dr beer wrote:Now I am not saying this has been done but in principle is there a flaw in this logic?


Yes, it lacks clear definitions for concepts like "extra linguistic capacity" and "physically possible". Consider a 1x1 square, the length of the diagonal is the square root of 2, does that make it "physically possible"? In fact, the ground you're touching on is very murky, and required many generations of mathematicians to fully understand and forms the foundation of real analysis. I recommend trying to understand the different numbers systems N, Z, Q, R and C particularly why and when we use all of them. This might help you to articulate your thoughts more clearly.
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Re: Math is a language, not a science?

Postby Dr beer » Sun Jul 14, 2013 5:14 pm UTC

Trust me I am aware exactly how murky this area is along with the nature and definitions of truth, reality and the relationship between language and physics (for example)
I have gone through the posts on this thread in some detail, many of which have been understood, many not and many seem resolved with a real world extension which kind of makes my point . If a scientist used math to predict the existence of a black hole, compound or element (for example) and then went on to locate it, or a philosopher used a word based language to deduct the existence of properties in the 'real world' and then found them, how would these statements be viewed? As arguments for the existence of whatever was being proposed they would certainly have integrity or 'truth' as it is generally understood and my initial question is no more complicated than this.

My take on what you guys have been discussing ( for what its worth) is that Math is to all intent and purpose a language, it is an invention not a discovery and its primary purpose is the description of the wider world (or you might as well be playing soduku). After all, are not all the symbols employed in math the product of someones imagination to describe functions and process ultimately in the physical world?
In categorising math as a language, would it therefore not be unreasonable to assume ( or explore the possibility of the fact ) that there MAY be axioms that apply universally to all language (s)? one of these being that if a concept has a real world extension, extra linguistic capacity etc then it has 'truth' or is 'proven' in the generally accepted meanings of these words ?

So...........

With the above in mind (as an example of where I am coming from) I would like to explore the duplicity of functions described by the symbols 'x' and '/' ( hereafter to represent division) in basic arithmatic. For example

Three guys have £5 each, in total they have £15 ( 3x5=15) a square measuring 1unit wide by 1unit long is 1unit sq .
In the first example we are doing no more than summing a number of same size sets whilst in the second example we are describing some kind of union or merger of two wholly distinct concepts resulting in a new concept.
If I highlight these differing functions by assigning differing symbols, for example when summing same size sets (general arithmatic multiplication) I use the symbol '@' and leave the symbol 'x' as is when used for describing physical process and we do the same for division, creating one symbol for the physical act of dividing and another for when we need to know how many same size sets are in a larger set ( ^ for the physical act and / to represent conventional division for example ) The upshot of this (apart from it being laughable to die hard mathematicians) is that the range of expression has been expanded whilst not affecting the result of any math/arithmatical working apart from in the area of Irrationals where you will find in some instances the language resulting in their existance ceases to exist and in others the question/sum itself becomes demonstratably irrational to start off with (ask a stupid question, get a stupid answer)

As will be obvious, my interest is more linguistic than math although I would like to use arithmatic, it symbols and the possiblity of eradicating certain irrationals to make a point elsewhere, which in turn takes me back to my original point. If language (math?) could be couched in such a way that a question/sum that had previously generated an irrational result now yielded a rational one or in the alternative, that the question itself was demonstratably irrational and therefore incapable of yielding a rational result, which would hold precedence?

( it is my understanding that irrationals are simply solutions to issues of consistency which I believe I deal with further down the line. Taking into account their lack of an extra linguistic capacity I am privately curious if their continued existence is necessary/important or even relevant but this is not what I am here to discuss. Hopefully you guys will be understanding of my lack knowledge in your field, find my curiosity non trival and in engage in dialogue, especially in the assigning of differing symbols to clearly differing process where I believe you will ultimately find my arguments have integrity and which I am happy to expand.)

One for the programmers I feel :twisted: :)

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Re: Math is a language, not a science?

Postby brenok » Sun Jul 14, 2013 6:16 pm UTC

I'm having problems understanding your post (which could be due to my lack of mathematical experience, English language or both). Why do you define two kinds of multiplication, and in which sense are you using the word "irrational"?

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Re: Math is a language, not a science?

Postby skeptical scientist » Mon Jul 15, 2013 5:06 am UTC

Dr beer wrote:Three guys have £5 each, in total they have £15 ( 3x5=15) a square measuring 1unit wide by 1unit long is 1unit sq .
In the first example we are doing no more than summing a number of same size sets whilst in the second example we are describing some kind of union or merger of two wholly distinct concepts resulting in a new concept.

These are really the same thing. If you have a rectangle measuring 3 units wide by 5 units long, it as 15 square units in area. Simultaneously, you can break it up into 3 rows of 5 one unit squares. So is the rectangle 3x5 units in area, or 3@5 units in area? How can it matter, when both 3x5 and 3@5 are just other names for the number 15?

The whole point of mathematics (and what makes it useful as a language) is to abstract away the differences between different physical objects and operations and discover and analyze the underlying unity in how they behave. So yes, mathematical objects are not completely real—they are abstracted from reality. This is not a defect, but an advantage.
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Re: Math is a language, not a science?

Postby eSOANEM » Mon Jul 15, 2013 9:21 pm UTC

brenok wrote:I'm having problems understanding your post (which could be due to my lack of mathematical experience, English language or both). Why do you define two kinds of multiplication, and in which sense are you using the word "irrational"?


In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)". It's a rather odd word and doesn't come from i-(rational) but rather from (i-ratio)-al where i- is one of the classical negating prefixes and -al an adjective-making one.
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Re: Math is a language, not a science?

Postby Sizik » Tue Jul 16, 2013 12:08 am UTC

skeptical scientist wrote:
Dr beer wrote:Three guys have £5 each, in total they have £15 ( 3x5=15) a square measuring 1unit wide by 1unit long is 1unit sq .
In the first example we are doing no more than summing a number of same size sets whilst in the second example we are describing some kind of union or merger of two wholly distinct concepts resulting in a new concept.

These are really the same thing. If you have a rectangle measuring 3 units wide by 5 units long, it as 15 square units in area. Simultaneously, you can break it up into 3 rows of 5 one unit squares. So is the rectangle 3x5 units in area, or 3@5 units in area? How can it matter, when both 3x5 and 3@5 are just other names for the number 15?


I would say that it would be 3x(5@1) or 5x(3@1), which would represent 3 rows of 5 or vice versa.
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Re: Math is a language, not a science?

Postby skeptical scientist » Tue Jul 16, 2013 1:33 am UTC

eSOANEM wrote:In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)". It's a rather odd word and doesn't come from i-(rational) but rather from (i-ratio)-al where i- is one of the classical negating prefixes and -al an adjective-making one.

Irrational does mean "not rational", so I'm not sure how you can say it doesn't come from i-rational. Rational itself comes from ratio+al, as it means "is a ratio (of integers)".
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Re: Math is a language, not a science?

Postby ConMan » Tue Jul 16, 2013 1:45 am UTC

I think he's trying to differentiate between rational = of a ratio, and rational = sensible, reasonable.
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Re: Math is a language, not a science?

Postby brenok » Tue Jul 16, 2013 4:45 am UTC

eSOANEM wrote:
brenok wrote:I'm having problems understanding your post (which could be due to my lack of mathematical experience, English language or both). Why do you define two kinds of multiplication, and in which sense are you using the word "irrational"?


In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)".


I know that. I was trying to figure out how does that make sense on context of this post.

Like:
Dr beer wrote:The upshot of this (apart from it being laughable to die hard mathematicians) is that the range of expression has been expanded whilst not affecting the result of any math/arithmatical working apart from in the area of Irrationals where you will find in some instances the language resulting in their existance ceases to exist and in others the question/sum itself becomes demonstratably irrational to start off with (ask a stupid question, get a stupid answer)

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Re: Math is a language, not a science?

Postby eSOANEM » Tue Jul 16, 2013 5:32 pm UTC

brenok wrote:
eSOANEM wrote:
brenok wrote:I'm having problems understanding your post (which could be due to my lack of mathematical experience, English language or both). Why do you define two kinds of multiplication, and in which sense are you using the word "irrational"?


In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)".


I know that. I was trying to figure out how does that make sense on context of this post.

Like:
Dr beer wrote:The upshot of this (apart from it being laughable to die hard mathematicians) is that the range of expression has been expanded whilst not affecting the result of any math/arithmatical working apart from in the area of Irrationals where you will find in some instances the language resulting in their existance ceases to exist and in others the question/sum itself becomes demonstratably irrational to start off with (ask a stupid question, get a stupid answer)


*Facepalm*

Serves me right for not double checking what post you were referring to. I'm pretty much as confused as you.

skeptical scientist wrote:
eSOANEM wrote:In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)". It's a rather odd word and doesn't come from i-(rational) but rather from (i-ratio)-al where i- is one of the classical negating prefixes and -al an adjective-making one.

Irrational does mean "not rational", so I'm not sure how you can say it doesn't come from i-rational. Rational itself comes from ratio+al, as it means "is a ratio (of integers)".


That's a good point. What I meant to say but what is not what I wrote was that it doesn't come from rational in the usual non-mathematical sense.
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Re: Math is a language, not a science?

Postby Tirian » Wed Jul 17, 2013 1:05 am UTC

eSOANEM wrote:That's a good point. What I meant to say but what is not what I wrote was that it doesn't come from rational in the usual non-mathematical sense.


The person or people who translated mathematical concepts into English were surely aware of the double entendre that rational numbers are a ratio of two natural numbers and also a set of numbers that are relatively closer to ordinary human intuition than their complement.

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Re: Math is a language, not a science?

Postby Copper Bezel » Wed Jul 17, 2013 2:03 am UTC

It's not a translation. It's an import, and both meanings existed in the same spelling in Latin. So far as I can tell, that second denotation of "outside reason" either developed again independently in English or was imported separately later on. Ask the friendly sphinx. Both are older in English than ratio, incidentally.

Someone with access to OED can check this one for certain.

The Latin sense of "thought" or "the sussing out" was first, though, and a ratio of numbers was a particular species of sussing for which the word apparently took on a technical sense. Either could have i- added on, to respectively create "without reason" and "unsussable." At that point, in applying the term to unsussable numbers, the dual meaning would have been clear, and it seems to have been fully intended, at least by some. (Don't tell me you hadn't heard that story?)
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Re: Math is a language, not a science?

Postby Tirian » Wed Jul 17, 2013 9:13 am UTC

Copper Bezel wrote:It's not a translation. It's an import, and both meanings existed in the same spelling in Latin.


I'm referring to the third and far more modern meaning, the one that describes a member of ℚ. Whoever did that (Dedekind?) had a choice of synonyms, and I would claim that the fact that the word "rational" is multifaceted would have been a specific point in its favor.

To give another example, consider the surreal numbers. Did Conway call them that because they are an extension of the real numbers, or did he call them that because their definition and utility is mind-blowing? Anyone who takes a single side of that debate needs to read more Conway.

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Re: Math is a language, not a science?

Postby Copper Bezel » Wed Jul 17, 2013 11:50 am UTC

I'm out of my depth, then, to understand the distinction you're making. I thought we were talking about the usual mathematical sense:
eSOANEM wrote:In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)".

Which is the same as the not-OED's "inexpressible in ordinary numbers", which is still the oldest usage of [ir]rational in English as imported from Latin (14th c.), while in Latin it was originally coined after the sense of "without reason" and was an intended "pun," if not a very funny one, for the Pythagoreans or some precursor.
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Re: Math is a language, not a science?

Postby gmalivuk » Wed Jul 17, 2013 7:30 pm UTC

Tirian wrote:I'm referring to the third and far more modern meaning, the one that describes a member of ℚ.
And that is different how, exactly, from the meaning that describes a number expressible as a ratio of two integers?
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Re: Math is a language, not a science?

Postby Tirian » Thu Jul 18, 2013 9:41 am UTC

The first two meanings of the word have synonyms, and the formal mathematical meaning does not. It was important that the term be chosen carefully, because you don't have any choice but to use that term when the context demands it.

To give an example of a poorly chosen term, whoever inflicted the term "vertical angles" on English-speaking humanity should be found and sternly spoken to, since it has nothing to do with vertical lines. On the other hand, rational numbers were named with care by someone or someones who chose against calling them reasonable numbers or fractional numbers. Because we have very little recourse now that those choices are made.

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Re: Math is a language, not a science?

Postby skeptical scientist » Thu Jul 18, 2013 2:36 pm UTC

Tirian wrote:The first two meanings of the word have synonyms, and the formal mathematical meaning does not. It was important that the term be chosen carefully, because you don't have any choice but to use that term when the context demands it.

Okay, I have no idea what you mean by "the first two meanings." I thought the first meaning was "amenable to reason" and the second was "member of Q," but apparently those aren't the two meanings you mean.
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Re: Math is a language, not a science?

Postby JainBug » Wed Jul 24, 2013 11:12 pm UTC

I think of math as a board game, where humans invented the rules. It still has real world equivalency.

The fundamental rules are, round about, those for counting, the rules for the signs, and addition and it's inverse. I believe you can derive multiplication and division from those. And then all the rest from that.

Math has rules, language does not, except grammar rules, but that is not the same kind of rule. E.g. math also has rules for it's visual representation.

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Re: Math is a language, not a science?

Postby Dr beer » Fri Jul 26, 2013 10:19 am UTC

skeptical scientist wrote:
Dr beer wrote:Three guys have £5 each, in total they have £15 ( 3x5=15) a square measuring 1unit wide by 1unit long is 1unit sq .
In the first example we are doing no more than summing a number of same size sets whilst in the second example we are describing some kind of union or merger of two wholly distinct concepts resulting in a new concept.

These are really the same thing. If you have a rectangle measuring 3 units wide by 5 units long, it as 15 square units in area. Simultaneously, you can break it up into 3 rows of 5 one unit squares. So is the rectangle 3x5 units in area, or 3@5 units in area? How can it matter, when both 3x5 and 3@5 are just other names for the number 15?

The whole point of mathematics (and what makes it useful as a language) is to abstract away the differences between different physical objects and operations and discover and analyze the underlying unity in how they behave. So yes, mathematical objects are not completely real—they are abstracted from reality. This is not a defect, but an advantage.


Actually I disagree with your first paragraph and will expand in my following post but I concur with your second point. My point (which fits into yours) was/is that if we visit the abstract dimension or world of imagination for whatever purpose be it fantasy, hope of scientific discoveries or physical inventions we at some point have to return or reference to the physical world for our abstractions to have 'truth'. Tying the properties of language (inc math) to the properties of physics sets parameters for a start... a linguistic statement or abstract that has no physical extension exists soley in the imagination, it has no 'truth'. Secondly, in a word is the object, object is the word type relationship language and subsequently perception become demonstratably subject to the laws and processes of physics and we can use the properties of language to deduce properties in physics/nature (this point also expanded on in my next post)

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Re: Math is a language, not a science?

Postby Dr beer » Fri Jul 26, 2013 12:29 pm UTC

eSOANEM wrote:
brenok wrote:I'm having problems understanding your post (which could be due to my lack of mathematical experience, English language or both). Why do you define two kinds of multiplication, and in which sense are you using the word "irrational"?


In the context of maths, irrational means "cannot be written in the form a/b where a and b are both integers (whole numbers)". It's a rather odd word and doesn't come from i-(rational) but rather from (i-ratio)-al where i- is one of the classical negating prefixes and -al an adjective-making one.


In reverse order, I am suggesting that irrational numbers fall into two camps, one is essentially arithmatic (namely sq root of 2 and n/0) the other is a comment/observation/reflection of physical property ( such as relationship between diagonal and length in a square, diameter and circumference and the golden ratio). All of which fall inline with your comment.

The premis of my original post was that if a linguistic statement does not have a real world or physical extention then it has no 'truth'( I am not claiming this as a profound point as it is how language is used intuitively and unconsciously by everybody, everyday.) yet irrational numbers would appear to fall into this hole.

Now keeping this premis to the forefront

A) n/0 is slightly different to the others so I deal with it seperately. A physical entity cannot be divided by a non physical entity, an integer or number referencing an event/entity in the physical dimension cannot therefore contain any amount of sets whose value is 0 or exists soley in the realm of abstraction so with regard to n/0=? a question that has no physical extension has been deemed to result in an answer which has no physical extension. The fact that both question and result exist soley in the imagination and in the absence of an axiom tying properties of language to properties of physics means that at present it can never be challenged. Tie language to physics and thats the first man down.

B) All my other examples of irrationals share a common quallity, namely that they seem to result in a series of never ending decimal places and cannot be expressed as a whole as you correctly pointed out. Referencing again my point about the relationship between abstract/physical and vice versa if we look at say the relationship between diameter and circumference and its 'irrational' numeric consequence we can quite simply deduce that there is no direct proportional relationship between the two and the best we can do is have an approximation. ( try and express numerically a ratio between two contsants that share no direct proportional relationship?) (where do even start?)

Moving on to my seperating the functions in arithmatic (defining two types of multiplication); that's not actually what I suggested
(disclaimer.. if I use excrutiatingly fascile examples I am not being patronising or condescending, the point I am trying to make is actually this simple but conversely (perversely) it is a very slippery concept to grasp ) and I did not limit it to just multiplication. Using your language, I seperated both multiplication and division with one half of each describing arithmatic (counting) type functions and the other physical process.

So, three cows in three fields how many cows? can be expresses 3x3=? this is in fact a linguistic representation of 111+111+111= with the answer being 111111111 or 9. The point I am repeating is in that the general arithmatic application of this symbol we are in fact doing no more than creating a shorthand for adding/summing a number of same value sets which will result in number summarising completely seperate, uniquely distinct events/entities.

When we get to the sum 1x1=1sq (1 pwr 11 etc) we are quite clearly reflecting on a union of the seperate qualities reflected in the question to create the combined quality of the result. eg L x W = sq , a square would not be a square if it did not posses both of the initial qualities, if the inital qualities could exist independent of each other there would be no square etc so who while under the traditional rules we can have 3x3=9 and 3x3=9sq the 9sq of the latter is actually ONE event to the value of 9sq....(massively different beasts and nowhere near the same in terms of result or implication (a nod to my previous post)

Getting to the point of seperating the two functions...
To all intent and purpose the basic arithmatic symbols are no more than arbitrary invention to describe function and process so for general arithmatic (three cows etc) I am now going to arbitrarily assign the symbol '@' and for geometry or describing/referencing physical process I am going to assign the symbol 'x' (h x w etc)

What I describe next is not a new argument but possibly what I draw down from it could possibly be considered novel.
A square root is the inverse of a square. Its essential quality is that as a set, the value within is equal to the number of times it fits into a whole ergo sq root of 16 is 4 (four sets of four)
starting at 1.. 1sq =1, 2sq =4 3sq =9 and so on. keeping in mind that root is inverse to sq, moving back down the line, sqr 0f 9 is 3, of 4 is 2, of 1 is 1. There is no sq root of 2. Expand this by examing the quality of 2 and you cannot get a number of sets whose value equal the divising number so in the ARITHMATIC capacity the square root of two does not exist either (second man down)

Relative to the relationship of a diagonal to length of a square I am not disputing that it shares the same numeric capacity as to what is commonly known as square root of two, but as we have now demonstrated that arithmatically, this does not actually exist so I arbitrarily assign it a name along the lines of Pi, for sake of argument 'Li' , its application/relevance however remains the same (would a rose smell so sweet by any other name :) )
.
The commom quality of Li, Pi, and the Golden Ratio is the never ending series of decimals so I just want to reiterate... these number do not exist in the physical realm, ergo they have no 'truth'. The clear inference from their existance is simply that there is no direct proportional representation or relativity between the constants bringing about their existance, this is not in itself 'irrational' its simply an aspect of physics and in abbreviating down these beasts to a just a few decimal places means we are reducing down to a rational aproximation for the purposes of language/expression/calculation etc, we are not utilising an irrational number, so in this capacity, irrational numbers simply highlights a relationship that does not exist/has no 'truth'. The term 'irrational number' is therefore to all intent and purpose a linguistic illusion, a mirage.... there is nothing there (the ratio's as approximations however are there ) (slippery)


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