I don't remember any philosophical threads about maths, especially regards its usefulness to philosophy. So...

The symbols employed by maths are all endowed with meaning. Whenever we read a mathematical equation, we know that each symbol has a meaning (for some symbols, this meaning also serves as a function/purpose). Fundamentally then, maths is just short-hand (or coded) everyday-language encompassing various concepts. The primary purpose of maths seems [to me] to be to seek the consequence/result of conjoining these concepts together, with a view to discovering something meaningful from the result.

Without doubt, maths has been of immense pragmatic value. The question is, of what value is maths when it comes to questions of, say, ontology or epistemology? We're all familiar with Zeno's paradox, for instance, but what are we to make of the 'proof' that motion is really possible via the conjunction of a selection of particular concepts to produce a result which ultimately means: "Yes, motion is really possible."? The bottom-line, in this case, is that the result unquestionably DOES claim to have something significant to say of ontological/metaphysical value and is also significant regards the epistemic reach of maths itself.

... However, as maths is nought but short-hand/code for everday language, one must undoubtedly question the conclusion derived from this proof, if not the result itself. For, can 'everyday language' establish that motion can be a real phenomenon, as opposed to just an experienced one? I don't think that it can, unless assumptions and/or mistakes are made in the lingual conctruction of the concepts/notions employed to construct such an argument. Certainly, it cannot be the case that maths draws conclusions of ontogical/metaphysical significance from symbols [of concepts] devoid of such values.

There are probably several other examples whereby maths makes conclusions of a philosophical significance. Cantor, for instance. The point is, are these conclusions justified? If not, then how can maths have any value for philosophy?

It might interest you to read a little about logicism, the theory championed by Russell and others, that some or all mathematics is reducible to logic. http://en.wikipedia.org/wiki/Logicism

What do you think of this theory? If you accept it, then would you say that logic has any value for philosophy?

jamest wrote:The symbols employed by maths are all endowed with meaning. Whenever we read a mathematical equation, we know that each symbol has a meaning (for some symbols, this meaning also serves as a function/purpose). Fundamentally then, maths is just short-hand (or coded) everyday-language encompassing various concepts.

I disagree even here. Mathematical symbols differ quite markedly from common-language in that they are rigidly defined. While everyday terms shift in meaning, the same does not hold true of mathematical symbols. At the same time mathematical concepts are thus limited by what we can rigidly define - everyday language allows for "soft symbols", which maths doesn´t.

jamest wrote:Without doubt, maths has been of immense pragmatic value. The question is, of what value is maths when it comes to questions of, say, ontology or epistemology?

That depends on how strongly you can formalize a particular view on ontology or epistemology. If you can get rid of soft symbols, a formal expression can help (it´s something I like to use when possible, mainly because it removes ambiguity, pretty much for the reasons given above).

susu.exp wrote:

jamest wrote:The symbols employed by maths are all endowed with meaning. Whenever we read a mathematical equation, we know that each symbol has a meaning (for some symbols, this meaning also serves as a function/purpose). Fundamentally then, maths is just short-hand (or coded) everyday-language encompassing various concepts.

I disagree even here. Mathematical symbols differ quite markedly from common-language in that they are rigidly defined. While everyday terms shift in meaning, the same does not hold true of mathematical symbols. At the same time mathematical concepts are thus limited by what we can rigidly define - everyday language allows for "soft symbols", which maths doesn´t.

That's all irrelevant, because the point is that mathematical symbols are loaded with meaning and that this meaning is derived from lingual concepts. Whether a concept is 'rigidly defined' has no significance whatsoever regards the point I was trying to make.

jamest wrote:Without doubt, maths has been of immense pragmatic value. The question is, of what value is maths when it comes to questions of, say, ontology or epistemology?

That depends on how strongly you can formalize a particular view on ontology or epistemology. If you can get rid of soft symbols, a formal expression can help (it´s something I like to use when possible, mainly because it removes ambiguity, pretty much for the reasons given above).

A concept [and it's corresponding symbol] does not automatically become of ontological/metaphysical value just because it's been rigidly defined.

Matthew Shute wrote:It might interest you to read a little about logicism, the theory championed by Russell and others, that some or all mathematics is reducible to logic. http://en.wikipedia.org/wiki/Logicism

What do you think of this theory? If you accept it, then would you say that logic has any value for philosophy?

It's a given that maths must be framed within logical principles. I'm not saying that maths isn't logical - I'm asking whether the results derived from mathematical endeavour facilitate the kind of metaphysical/ontological/epistemological conclusions which are sometimes inferred from said results. If not, then maths is not an endeavour that is actually useful to philosophy.

jamest wrote:That's all irrelevant, because the point is that mathematical symbols are loaded with meaning and that this meaning is derived from lingual concepts. Whether a concept is 'rigidly defined' has no significance whatsoever regards the point I was trying to make.

Can you provide an example? I would debate that the meaning of mathematical symbols is neccessarily derived from lingual concepts. For instance there are issues with common language uses of infinity that arise because a formal definition of infinity leads to the conclusion that not all infinities are equally large - a distinction that isn´t derrived from common language notions of infinity.

jamest wrote:A concept [and it's corresponding symbol] does not automatically become of ontological/metaphysical value just because it's been rigidly defined.

You´ve got my point backwards. If a concept can be rigidly defined we can perform formal operations. If we perform formal operations, we are doing maths. So if a concept is useful within metaphysics and it can be rigidly defined, then maths can be used to operate with this concept and it is useful for the metaphysics.

There goes LF, derailing this thread into infinity.

"Infinity equates to an indivisible oneness" isn´t mathematics. If had to find a sentence to illustrate what I mean by "soft symbols", this´d be a good one to start with.
The size of a set S is infinite iff there is a true subset S', so that there´s a bijective relation f:S->S'.
Equality is what you get when you take an equivalence relation ~ defined on a set U and move to a set U* of equivalence classes through a function g:U->U* with x~y => g(x)=g(y)
1 indicates the neutral element regarding the multiplication operation in a ring.
A number n is divisible by another number m is n MOD m=0.
String these rigid definitions together as in the above sentence and you get BS. If that sentence means anything, it´s nothing that uses the formal language of maths.

susu.exp wrote:

jamest wrote:That's all irrelevant, because the point is that mathematical symbols are loaded with meaning and that this meaning is derived from lingual concepts. Whether a concept is 'rigidly defined' has no significance whatsoever regards the point I was trying to make.

Can you provide an example? I would debate that the meaning of mathematical symbols is neccessarily derived from lingual concepts. For instance there are issues with common language uses of infinity that arise because a formal definition of infinity leads to the conclusion that not all infinities are equally large - a distinction that isn´t derrived from common language notions of infinity.

Hold on, Sir. The conclusion that "not all infinities are equally large" is derived from the results of a mathematical endeavour which first derived the meaning of infinity from a purely lingual basis. What you say, then, does not refute the notion that mathematical symbols (of infinity, in this instance) are mere representations of pre-constructed lingual concepts.

I acknowledge that there are philosophical issues with the definition of infinity. I, for one - as LF just noted - am one of those people who do not accept the 'rigid' definition of infinity (adopted by maths) as being synonymous with the rational definition of infinity. In fact, this acts as a case-in-point, since I would argue that the rigid definition of infinity is sufficiently irrational so as to prohibit ontological/metaphysical/epistemological conclusions being drawn from the results of any mathematical endeavour that had utilised said concept.

You seem to be turning everything upon its head, here. The conclusion that "not all infinities are equally large", only applies to those particular pre-defined infinities - which may not bear any resemblance to rational and/or 'real' infinities (if indeed 'infinity' could even mean something within actual existence). Therefore, how could the results derived from such maths have any kind of philosophical value?

jamest wrote:A concept [and it's corresponding symbol] does not automatically become of ontological/metaphysical value just because it's been rigidly defined.

You´ve got my point backwards. If a concept can be rigidly defined we can perform formal operations. If we perform formal operations, we are doing maths.

Okay, but...

... So if a concept is useful within metaphysics and it can be rigidly defined, then maths can be used to operate with this concept and it is useful for the metaphysics.

Such a stance is a hindrance to metaphysical debate, of course, because if one rigidly defines a concept to mean 'X' and maths operates within this rigid definition, then maths can draw nought but conclusions commensurate with this pre-determined definition of X. In a nutshell, any such conclusion would be self-serving and ultimately circular.

jamest wrote:Hold on, Sir. The conclusion that "not all infinities are equally large" is derived from the results of a mathematical endeavour which first derived the meaning of infinity from a purely lingual basis. What you say, then, does not refute the notion that mathematical symbols (of infinity, in this instance) are mere representations of pre-constructed lingual concepts.

No, it doesn´t. It defines infinity through the property of some operations. I think the only primitive notions used there are sets and relations - anything else is defined in terms of them. But while we´re at it: What is the pre-constructed lingual concept behind Non-Borellian sets on [0,1]? Can you describe one without using formal operations?

jamest wrote:I acknowledge that there are philosophical issues with the definition of infinity. I, for one - as LF just noted - am one of those people who do not accept the 'rigid' definition of infinity (adopted by maths) as being synonymous with the rational definition of infinity. In fact, this acts as a case-in-point, since I would argue that the rigid definition of infinity is sufficiently irrational so as to prohibit ontological/metaphysical/epistemological conclusions being drawn from the results of any mathematical endeavour that had utilised said concept.

What in particular do you hold to be irrational about the definition?

jamest wrote:You seem to be turning everything upon its head, here. The conclusion that "not all infinities are equally large", only applies to those particular pre-defined infinities - which may not bear any resemblance to rational and/or 'real' infinities (if indeed 'infinity' could even mean something within actual existence). Therefore, how could the results derived from such maths have any kind of philosophical value?

Can you formalize your concept of infinity? If you can (and the result is consistent) you can construct a jamestian set theory. It´d still be maths. If you can´t formalize it this means that your notion of infinity remains ambiguous and this in turn might make it hard to communicate. The job of maths as a language is to reduce ambiguity and thus allow for a less error-prone communication.

jamest wrote:Such a stance is a hindrance to metaphysical debate, of course, because if one rigidly defines a concept to mean 'X' and maths operates within this rigid definition, then maths can draw nought but conclusions commensurate with this pre-determined definition of X. In a nutshell, any such conclusion would be self-serving and ultimately circular.

Any metaphysical debate ultimately has to rest on axioms and axioms define the basic terms. Metaphysics has to define reality and as a result the only conclusions you can reach are commensurate with this definition. I don´t see how this problem is evaded by using ambiguous common language gratuitously: If we have a common understanding of what the word apple means, we can draw no conclusion that an apple is not an apple (provided we stick to logics). If you can prove the negation of a statement from the statement that statement is absurd. If your criticism of mathematics resides in the finding that mathematics is a non-absurd enterprise, I think it will wither it...

all of mathematical philosophy can be reduced to pondering over finte,infinite and sunyata(void) and trying to work out relations between them.

susu.exp wrote:Can you formalize your concept of infinity? If you can (and the result is consistent) you can construct a jamestian set theory. It´d still be maths. If you can´t formalize it this means that your notion of infinity remains ambiguous and this in turn might make it hard to communicate. The job of maths as a language is to reduce ambiguity and thus allow for a less error-prone communication.

This could be why jamest resists formalising jamestian infinity, all the while arguing that mathematics is useless in philosophy. The jamestian infinity is an important part of jamest's philosophy, it seems - and jamestian philosophy has to be sound... he's already "worked it out." You do the maths.

Jamestian infinity is not math but is analogous to that blowing sound you hear whales make. It's more of a behavior of the wild type. You don't define this, you just Observe it whilst being careful nought to mistake it for the thing itself.

jamest wrote:

susu.exp wrote:

jamest wrote:That's all irrelevant, because the point is that mathematical symbols are loaded with meaning and that this meaning is derived from lingual concepts. Whether a concept is 'rigidly defined' has no significance whatsoever regards the point I was trying to make.

Can you provide an example? I would debate that the meaning of mathematical symbols is neccessarily derived from lingual concepts. For instance there are issues with common language uses of infinity that arise because a formal definition of infinity leads to the conclusion that not all infinities are equally large - a distinction that isn´t derrived from common language notions of infinity.

Hold on, Sir. The conclusion that "not all infinities are equally large" is derived from the results of a mathematical endeavour which first derived the meaning of infinity from a purely lingual basis. What you say, then, does not refute the notion that mathematical symbols (of infinity, in this instance) are mere representations of pre-constructed lingual concepts.

I acknowledge that there are philosophical issues with the definition of infinity. I, for one - as LF just noted - am one of those people who do not accept the 'rigid' definition of infinity (adopted by maths) as being synonymous with the rational definition of infinity. In fact, this acts as a case-in-point, since I would argue that the rigid definition of infinity is sufficiently irrational so as to prohibit ontological/metaphysical/epistemological conclusions being drawn from the results of any mathematical endeavour that had utilised said concept.

You seem to be turning everything upon its head, here. The conclusion that "not all infinities are equally large", only applies to those particular pre-defined infinities - which may not bear any resemblance to rational and/or 'real' infinities (if indeed 'infinity' could even mean something within actual existence). Therefore, how could the results derived from such maths have any kind of philosophical value?

jamest wrote:A concept [and it's corresponding symbol] does not automatically become of ontological/metaphysical value just because it's been rigidly defined.

You´ve got my point backwards. If a concept can be rigidly defined we can perform formal operations. If we perform formal operations, we are doing maths.

Okay, but...

... So if a concept is useful within metaphysics and it can be rigidly defined, then maths can be used to operate with this concept and it is useful for the metaphysics.

Such a stance is a hindrance to metaphysical debate, of course, because if one rigidly defines a concept to mean 'X' and maths operates within this rigid definition, then maths can draw nought but conclusions commensurate with this pre-determined definition of X. In a nutshell, any such conclusion would be self-serving and ultimately circular.

Hmm. Why would one wish to have the freedom in mathematics to draw conclusions that are not necessarily commensurate with pre-determined, hard definitional concepts like addition? Would we like the use of addition to end differently, to conclude that 2+2 equals 3 or 5? I don't think the conclusions are circular as much as they are consistent and determinable.

susu.exp wrote:

jamest wrote:Hold on, Sir. The conclusion that "not all infinities are equally large" is derived from the results of a mathematical endeavour which first derived the meaning of infinity from a purely lingual basis. What you say, then, does not refute the notion that mathematical symbols (of infinity, in this instance) are mere representations of pre-constructed lingual concepts.

No, it doesn´t. It defines infinity through the property of some operations.

No it doesn't. The concept of infinity drawn from the results of such operations is merely borrowed/drawn from philosophical definitions of infinity which were formulated (lingually constructed) well in excess of 2000 years-ago. Modern mathematicians did not invent the concept of infinity.

I think the only primitive notions used there are sets and relations - anything else is defined in terms of them.

That's incorrect, as explained.

But while we´re at it: What is the pre-constructed lingual concept behind Non-Borellian sets on [0,1]? Can you describe one without using formal operations?

You are too hung-up on formality and rigidity. As I explained in an earlier post, this is irrlevant regards the point I'm trying to make.

jamest wrote:I acknowledge that there are philosophical issues with the definition of infinity. I, for one - as LF just noted - am one of those people who do not accept the 'rigid' definition of infinity (adopted by maths) as being synonymous with the rational definition of infinity. In fact, this acts as a case-in-point, since I would argue that the rigid definition of infinity is sufficiently irrational so as to prohibit ontological/metaphysical/epistemological conclusions being drawn from the results of any mathematical endeavour that had utilised said concept.

What in particular do you hold to be irrational about the definition?

The notion that infinity is a set of finite things, yet transcends finiteness. Ultimately, we must relate this folly to existence in order to see the irrationality of it all. At this juncture - early in the thread - I'm reluctant to move the focus of this discussion.

jamest wrote:You seem to be turning everything upon its head, here. The conclusion that "not all infinities are equally large", only applies to those particular pre-defined infinities - which may not bear any resemblance to rational and/or 'real' infinities (if indeed 'infinity' could even mean something within actual existence). Therefore, how could the results derived from such maths have any kind of philosophical value?

Can you formalize your concept of infinity? If you can (and the result is consistent) you can construct a jamestian set theory.

What do you mean by 'formalize'? I can provide a rational definition of infinity, but it won't be a mathematical presentation. In fact, there are no existing mathematical concepts which would facilitate this.

It´d still be maths. If you can´t formalize it this means that your notion of infinity remains ambiguous and this in turn might make it hard to communicate. The job of maths as a language is to reduce ambiguity and thus allow for a less error-prone communication.

Less error-prone? My whole point is that the philosophical conclusions drawn from maths are utterly error-ridden. Remember, the ultimate focus of this thread is philosophy, not maths.

THWOTH wrote:
Hmm. Why would one wish to have the freedom in mathematics to draw conclusions that are not necessarily commensurate with pre-determined, hard definitional concepts like addition? Would we like the use of addition to end differently, to conclude that 2+2 equals 3 or 5? I don't think the conclusions are circular as much as they are consistent and determinable.

I'm talking about drawing significant philosophical conclusions from mathematical results. The results themselves are sound in relation to the axioms which spawn them.