Is pure mathematics the basis for all thought?

[Functor97]

I don't know what you mean by "our physics" so I'll interpret your position thusly:

Mathematics is process of converting axoimatic categories into statements via a logic system, therefore all thought, which can be reduced to this process, is mathematics.

Is this correct? If so, then I still can't agree. As I said before, humans change their axioms and their logic systems based on context. That process is outside mathematics itself.



I have no problem with intuitionistic mathematics being mathematics, I have a problem with it being called mathematical logic, which to me means first-order logic because that is what I use. It's like getting a chemist to admit chemistry is actually physics, it may be right but it you won't get anywhere.

You keep changing your position, or so it seems to me. Yes our axioms are arbitary, and yes the process of reasoning from those axioms (which see as mathematics) remains the same. I have been saying that all along, it is not so much content as process which matters. Our changing of logical tautology is outside of mathematics, i agree, it is logic, when we apply those new rules that becomes mathematics. So in a way logic/philosophy is the basis of all thought, and from mathematics it goes onto physics and so on, from my reductionist perspective.

 

[pwsnafu]

You keep changing your position, or so it seems to me. Yes our axioms are arbitary, and yes the process of reasoning from those axioms (which see as mathematics) remains the same. I have been saying that all along, it is not so much content as process which matters. Our changing of logical tautology is outside of mathematics, i agree, it is logic, when we apply those new rules that becomes mathematics. So in a way logic/philosophy is the basis of all thought, and from mathematics it goes onto physics and so on, from my reductionist perspective.

It's occurred to me that my most important assumption has been unstated. Mathematics is not the same as philosophy. In the former all definitions are well-defined (no pun intended). A vector space is well-defined construct, it doesn't matter if you are doing analysis or topology. But in philosophy, this is not true. http://www.smbc-comics.com/index.php?db=comics&id=1879#comic" did a humorous take on this.
Now both math and philosophy are applied logic. If you include fuzzy logic and its cousins then you can deal with vagueness in truth as well. But you can't deal with vagueness of definitions. That is not mathematics.

I keep bringing up changing changing axioms and logic systems. Humans are able to have deductions which change the definitions as the discourse changes the context, and this happens in philosophy papers. The concept of "deity" can mean one thing in one section and another in a second section. That would be a disaster in mathematics. Indeed Newton, when doing calculus, was criticized many times for defining h to be non-zero, and then letting h to be zero after division. This type of thing is allowed in philosophy.

Now consider this question: "Is intuitionistic logic better than first-order logic?" I'm taking two logic systems and comparing them. But the quantifier "better" is ill-defined. This is not a mathematics problem. It's philosophy. "Should I use intutionistic logic and not first-order logic for this problem?" is also not a mathematical question. Switching from one logic system to another is not a mathematical process.

 

[Functor97]

It's occurred to me that my most important assumption has been unstated. Mathematics is not the same as philosophy. In the former all definitions are well-defined (no pun intended). A vector space is well-defined construct, it doesn't matter if you are doing analysis or topology. But in philosophy, this is not true. http://www.smbc-comics.com/index.php?db=comics&id=1879#comic" did a humorous take on this.
Now both math and philosophy are applied logic. If you include fuzzy logic and its cousins then you can deal with vagueness in truth as well. But you can't deal with vagueness of definitions. That is not mathematics.

I keep bringing up changing changing axioms and logic systems. Humans are able to have deductions which change the definitions as the discourse changes the context, and this happens in philosophy papers. The concept of "deity" can mean one thing in one section and another in a second section. That would be a disaster in mathematics. Indeed Newton, when doing calculus, was criticized many times for defining h to be non-zero, and then letting h to be zero after division. This type of thing is allowed in philosophy.

Now consider this question: "Is intuitionistic logic better than first-order logic?" I'm taking two logic systems and comparing them. But the quantifier "better" is ill-defined. This is not a mathematics problem. It's philosophy. "Should I use intutionistic logic and not first-order logic for this problem?" is also not a mathematical question. Switching from one logic system to another is not a mathematical process.

I will concede i was wrong to claim that everything was mathematics before. Would it be correct to say that philosophy sets the axioms and mathematics applies them? Deciding upon the logical qualifiers is in my opinion philosophy, creating the basic rules of a logic system then too must be philosophy. I guess this begs the question are pure mathematicians just applied philosophers? In my experience mathematicians and scientists often criticize philosophy, and portray it as pointless/useless and at odds with the scientific method. I do not like this model, all science being based upon philosophy, but it makes the most sense.
When i was young i thought of mathematics in a platonic sense. It seems the more mathematics i study the less and less sure i become of its perfection

 

[Studiot]

I will concede i was wrong to claim that everything was mathematics before.

Which was the point of my earlier posts. Some, yes, even much, perhaps most, but everything (all) no.

 

[disregardthat]

. Indeed Newton, when doing calculus, was criticized many times for defining h to be non-zero, and then letting h to be zero after division. This type of thing is allowed in philosophy.

There is nothing wrong with what Newton did other than the bad pedagogical effect.

Creating a system in which he uses a constant, say h, such that h^2 = 0, but h =/= 0 is perfectly fine. While I'm not well-versed in algebraic geometry, I believe this is very close to what is going on when calculating derivatives of algebraic curves. It can be made "rigorous" (consider R[x]/x^2), but that's not the point. Simply stating and using the rules and possible operations is still mathematics, regardless of having a (rigorous or not) definition behind it with respect to some axiomatic system.

The main point here is that it cannot be criticized for not being mathematics, only for being "mysterious", counterintuitive (or ghosts of departed quantities) etc which are not mathematical objections.

 

[SteveL27]

There is nothing wrong with what Newton did other than the bad pedagogical effect.

Creating a system in which he uses a constant, say h, such that h^2 = 0, but h =/= 0 is perfectly fine. While I'm not well-versed in algebraic geometry, I believe this is very close to what is going on when calculating derivatives of algebraic curves. It can be made "rigorous" (consider R[x]/x^2), but that's not the point. Simply stating and using the rules and possible operations is still mathematics, regardless of having a (rigorous or not) definition behind it with respect to some axiomatic system.

The main point here is that it cannot be criticized for not being mathematics, only for being "mysterious", counterintuitive (or ghosts of departed quantities) etc which are not mathematical objections.

And what are these fluxions? The velocities of evanescent increments. And what are these
evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we call them
ghosts of departed quantities? … He who can digest a second and third fluxion, a second or third difference, need not,
methinks, be squeamish about any point in Divinity.

– George Berkeley

Point being that Newton's work was not rigorous; Berkeley (pronounced "Barkley," like the basketball player) knew it wasn't rigorous; and Newton's own struggles over the years to reformulate his use of infinitesimals shows that even Newton knew his work wasn't rigorous.

Inventing a method that works, and showing that the method is logically sound, are two different things. Of course Newton was a great mathematician, but let's not confuse greatness with logical soundness.

 

[disregardthat]

Inventing a method that works, and showing that the method is logically sound, are two different things. Of course Newton was a great mathematician, but let's not confuse greatness with logical soundness.

The problem is that there is no way of showing that such a method (or any method relying on basic arithmetic) is logically sound whatsoever, as Gödel has proved. Berkeley's objections would have had mathematical relevance if he had pointed out contradictions, errors, but not if they were on the basis of mistrust of the soundness of Newton's methods. At any point we may find contradictions in our methods, but that just calls for a slight change to prevent them (e.g. naive set theory). There is nothing wrong about utilizing a mathematical method just because one is not confident in its logical soundness. At all times we employ this method of working; creating mathematical rules to utilize without being certain as to whether we will fall into contradiction (we don't know whether set theory is consistent or not).

There is a problem and a lot of confusion about the notion of "rigour". One will have difficulty defining this for mathematics, even though we easily say that some things are rigorous while other things are not. It is in fact a question of the degree of confidence.

 

[SteveL27]

There is nothing wrong about utilizing a mathematical method just because one is not confident in its logical soundness.

That is the same point I was trying to make, though perhaps not well enough to be clear.

Newton's own attempts over the years to rework infinitesimals in various ways, show that he well understood the distinction between effectiveness and soundness.

I agree with you that it's perfectly ok to use techniques that work; and allow the soundness to be worked out later. (In the case of Newton's calculus, that process took around 200 years!)

But one should never say that because a technique works, that therefore it is sound.

Re-reading your post, I think we were always in agreement on that point.

 

[Studiot]

But one should never say that because a technique works, that therefore it is sound

I like Heaviside's comment on this.

 

[Functor97]

The problem is that there is no way of showing that such a method (or any method relying on basic arithmetic) is logically sound whatsoever, as Gödel has proved. Berkeley's objections would have had mathematical relevance if he had pointed out contradictions, errors, but not if they were on the basis of mistrust of the soundness of Newton's methods. At any point we may find contradictions in our methods, but that just calls for a slight change to prevent them (e.g. naive set theory). There is nothing wrong about utilizing a mathematical method just because one is not confident in its logical soundness. At all times we employ this method of working; creating mathematical rules to utilize without being certain as to whether we will fall into contradiction (we don't know whether set theory is consistent or not).

There is a problem and a lot of confusion about the notion of "rigour". One will have difficulty defining this for mathematics, even though we easily say that some things are rigorous while other things are not. It is in fact a question of the degree of confidence.

Very interesting. Are you claiming that mathematics is in essence a calculating game? Your interpretation would make mathematics no different to physics, which is often disparaged for lacking rigor.
I guess my question is, can mathematics ever be perfectly rigorous?

 

[chiro]

Very interesting. Are you claiming that mathematics is in essence a calculating game? Your interpretation would make mathematics no different to physics, which is often disparaged for lacking rigor.
I guess my question is, can mathematics ever be perfectly rigorous?

Mathematics is a language, and like any language it is evolving.

If inconsistencies are found in math, like any language, it needs to go through reformulation.

This happens with every language.

There is an important facet though of mathematics that is somewhat paradoxical: mathematics is able to be so broad, yet so precise. This kind of property makes it a great language as not many languages have this property.

If our descriptive capacity is lacking to consistently describe something, we will ultimately have to create lingual definitions that fill the gap: this is what has happened before and I don't see it stopping anytime soon.