Is pure mathematics the basis for all thought?

[Studiot]

Even when you do it in your mind..

One of the points I am trying to make is that the whole of say the x-y plane is still there whether you use it or not in any particular graph.

 

[Functor97]

Even when you do it in your mind..

An old philosopher joke
If you conceive nothing: Do you still exist?

I have had to change my view of mathematics to a form of : studies of quantity and the relations there of.

I went through what archeologist have found, and some of what anthropologist have found.

The development of quantity by man has gone through changes to what is and is not accepted. It even took the old philosophers games and rules of argument and mathematics and others to formalism it into logic of argument. Even today there are different forms of such that allow different trains of thought to what is and is not accepted as proof, not just in mathematics.

I have had to discard the "purity" idea of such. If so it could easily be used to communicate with other species of our world. Some times it is difficult to use it to communicate with it to other people. I would say it is not so 'pure".

That I right or wrong, Who is to say. Not me.
Yet varied questions keep pooping up by folks thinking about careers, questions of why math unto its self, works for other science etc... The ideas of both science and mathematics having the potential not to be so exact, etc.

Back to the original subject. :

I have also found across the years people searching for the ideas of a set system, to go into fields such as science an math, and others.

I have encountered folks that leave physics when they hit the uncertainty principle.

Or math due to the original posters statement of
"Finally if mathematics is just axioms, and we cannot prove an axiom to be true, and yet mathematics is the basis of all science, does this mean absolute truth is beyond us?"

More often than not: (The idea the world around us is not so ridged, that it is not an exact) seems to be the root of the conflict and confusion.
Of course the root of the question is way deeper than just the math or science. It seems to bring up questions in the root of the person them selves. It creates an internal crisis for the individual, and the lack of "truth", "Purity", and the result of the idea of "control".

To be honest I have not found a good way to talk to and or see a person, who would be good and or great in these fields: to continue past the self conflict encountered with these questions and problems. To many times I see people just stopping and going different routes due not
to the questions asked here, but the internal conflict they create in them selves from the questions. I have no doubt that a great many talents to math and science, are lost in this manner.

Very interesting, but surely you agree mathematics, or physics at the least, is as close to eternal truth as we will ever come? Surely its the journey not the destination.

 

[Functor97]

'helped' is a long way from 'forms the basis of'.

I did ask if you considered logic part of pure maths and the question has again be raised by
disregardthat.

I agree the thread would benefit from firming up on definitions.

A similar question is also current here and the references in post 21 and 31 are qite relevant.

https://www.physicsforums.com/showthread.php?t=511119

go well

You see that thread highlights exactly what i have been trying to get at. The old tree in the forest argument aside, we are now dealing with the sum of histories rather then exact Newtonian events, which is in essence pure mathematics. This has gotten a lot deeper then just the uncertainty principle, we now see our physical objects as mathematical ones, so why is physics not just a subset of mathematics?

 

[Studiot]

I (we) still don't know if you are including logic within mathematics.
The answer to this makes a considerable difference.

Consider this sentence.

It was raining so I took my umbrella in order to remain dry.

I see logic in play here , but personally I see no mathematics. Amongst other ideas, the sentence contains the concept of a goal which I believe to be a thought process, independent of mathematics.


Your original thesis was 'basis for all thought'

You started (rightly) with the technical also made an attempt to extend to the arts which is understandable considering the all emcompasing nature of your thesis.

My comment about cooking was prompted by this arty dimension.

As regards the lesser question "Is Physics a subset of Mathematics?"
I am not sure. I like to have explanations, separate from the mathematics, for physical processes, but it is hard to imagine Physics totally without mathematics.

go well

 

[Functor97]

I (we) still don't know if you are including logic within mathematics.
The answer to this makes a considerable difference.

Consider this sentence.

It was raining so I took my umbrella in order to remain dry.

I see logic in play here , but personally I see no mathematics. Amongst other ideas, the sentence contains the concept of a goal which I believe to be a thought process, independent of mathematics.


Your original thesis was 'basis for all thought'

You started (rightly) with the technical also made an attempt to extend to the arts which is understandable considering the all emcompasing nature of your thesis.

My comment about cooking was prompted by this arty dimension.

As regards the lesser question "Is Physics a subset of Mathematics?"
I am not sure. I like to have explanations, separate from the mathematics, for physical processes, but it is hard to imagine Physics totally without mathematics.

go well

I would judge mathematics to be applied logic, but we cannot have an axiomatic logic system which is consistent and perfect, so i would judge the axioms of mathematics to be the "arbitary" ones of logic.

As for physics containing physical reasoning, i am really questioning if you have studied much/any modern physics. The whole reason relativity and QM came as such as shock was that we could not use our ordinary intuition, the answers become mathematical not physical. Unless physics gradually evolves to become more "realistic" with guided mathematics, i think we have simply uncovered the fact that reality, as far as our human brains can ever percieve is mathematical. Furthermore i think all of art has its basis in mathematics, not simply drawing pretty elucidean solids, but the abstract ideas behind the artwork (the reason we create art) is mathematical, deductive, but arbitary.

 

[Studiot]

This is a nice friendly discussion to which I hope I am contributing something worthwhile.

So please address my points as well as simply restating your own.

If logic is a subset of maths then maths includes examples like my umbrella.

If maths is a subset of logic then there are strands of thought outside maths.

I would like to take may culinery example further as well.
I know there are some who advocate a mathematical basis for graphic arts and music so I avoided that area, but there are also some who consider 'culinery arts'.

go well

 

[Functor97]

This is a nice friendly discussion to which I hope I am contributing something worthwhile.

So please address my points as well as simply restating your own.

If logic is a subset of maths then maths includes examples like my umbrella.

If maths is a subset of logic then there are strands of thought outside maths.

I would like to take may culinery example further as well.
I know there are some who advocate a mathematical basis for graphic arts and music so I avoided that area, but there are also some who consider 'culinery arts'.

go well

I am sorry if i came off as a bit rude.

Mathematics is a subset of logic, in my view logic and mathematics have a very vague separation at the elementary level, i.e. when have i stopped doing logic, but have moved onto mathematics? I do not think your claim that mathematics being a subset of logic implies other subjects. Logic is the very basis for mathematics, mathematics in turn applies that logic to models. Chemistry, Physics, History, Business, Law, art, music all exist, i do not deny that, but their basis is mathematical. Think of logic being the peak of an inverted pyramid, with mathematics covering the top but extending further, As you go further up the pyramid the mathematics becomes more and more arbitary, and the applications are evident. Like i said, mathematics is applied logic.

I do not know if i can claim all parts of the arts are a part of mathematics, maybe it is only formal knowledge that is mathematical, i am not sure. The ideas, or feelings expressed in the art are abstract, and thus are mathematical in my view, they may not be rigorous, but they use a vague intuition.

Think of our brains as a computer, mathematics is our source code and thus we view reality as a part of mathematics. If it really is or not, we will never know. We may evolve further, or come in contant with more intelligent life forms, but if the latter does exist, it is not evident that they will be able to communicate their "science" to us, no more than we can to our pet dogs.

 

[Studiot]

Not rude, but I put up the examples for a reason.

OK so can we move on from all thought to some thought?

 

[lostcauses10x]

Very interesting, but surely you agree mathematics, or physics at the least, is as close to eternal truth as we will ever come? Surely its the journey not the destination.

Well I do not see it as "eternal truth". more an internal understanding of external input that can be repeated, therefore the ideas of patterns happen.

As for the journey to were we are today?

So far these patterns hold useful to the the world and universe we exist in at this time. We can speculate they are truths that may hold to the entirety of the universe, yet who is to say such is "truth", or "absolute".
Note this also depends on how an individual defines the terms "truth", and or "absolute".

I see mathematics as the quantity: (as statement, without a noun, or qualifier.) ; and the relations that are found in the changes of quantity.
Strangely enough so far, mankind is the only thing that seems to use mathematics in our world, and so far the universe; as a form of communication. Even with man it has not and is not so universal.

I see physics as the observations of our surrounding world in which we have managed to recognize a pattern, and is steady enough to be able to measure. From that form some of the rules have been steady enough to allow man to actually build ideas of our physical world,and machines.
This actually comes from the measurement of such physical relations. Of course measurement is a quantity with a qualifier, or noun attached.

To be honest for man to have been able to communicate such ideas in the past forward (memory, even the written language is a form of such) to the extent we can build a car, radio, computer, power plant, etc. To me is a almost unbelievable achievement.

I also see math as the primary tool of communicating the changes we observe in physics.
This is due to "quantity", and the relations of such changes in quantity we observe and conceive; we also in reality as man studied the changes in quantity, which is what we now call mathematics. These readily found use in physics due to it was already a language of change.

As for logic, it was formed out of argument, and of course adapted to math, and even the sciences. To be honest it is in my opinion, logic is just another set of rules that formed out of the necessity of mans social groups getting larger and larger. A process of argument had to be formed, or results would not be able to happen. For mankind to be able to do such again is an almost unbelievable achievement.

We did not come to were we are today easily. Mankind's ideas and belief founded upon such ideas, has; and may still tend to hinder what mankind can do. We tend to suppress, damage, and or kill them that do not follow the norm.

 

[Functor97]

Not rude, but I put up the examples for a reason.

OK so can we move on from all thought to some thought?

Like i said, i think all thought is mathematical, based upon logical axioms.

 

[Functor97]

Well I do not see it as "eternal truth". more an internal understanding of external input that can be repeated, therefore the ideas of patterns happen.

As for the journey to were we are today?

So far these patterns hold useful to the the world and universe we exist in at this time. We can speculate they are truths that may hold to the entirety of the universe, yet who is to say such is "truth", or "absolute".
Note this also depends on how an individual defines the terms "truth", and or "absolute".

I see mathematics as the quantity: (as statement, without a noun, or qualifier.) ; and the relations that are found in the changes of quantity.
Strangely enough so far, mankind is the only thing that seems to use mathematics in our world, and so far the universe; as a form of communication. Even with man it has not and is not so universal.

I see physics as the observations of our surrounding world in which we have managed to recognize a pattern, and is steady enough to be able to measure. From that form some of the rules have been steady enough to allow man to actually build ideas of our physical world,and machines.
This actually comes from the measurement of such physical relations. Of course measurement is a quantity with a qualifier, or noun attached.

To be honest for man to have been able to communicate such ideas in the past forward (memory, even the written language is a form of such) to the extent we can build a car, radio, computer, power plant, etc. To me is a almost unbelievable achievement.

I also see math as the primary tool of communicating the changes we observe in physics.
This is due to "quantity", and the relations of such changes in quantity we observe and conceive; we also in reality as man studied the changes in quantity, which is what we now call mathematics. These readily found use in physics due to it was already a language of change.

As for logic, it was formed out of argument, and of course adapted to math, and even the sciences. To be honest it is in my opinion, logic is just another set of rules that formed out of the necessity of mans social groups getting larger and larger. A process of argument had to be formed, or results would not be able to happen. For mankind to be able to do such again is an almost unbelievable achievement.

We did not come to were we are today easily. Mankind's ideas and belief founded upon such ideas, has; and may still tend to hinder what mankind can do. We tend to suppress, damage, and or kill them that do not follow the norm.

The interesting thing is, why only one logic? We cannot have two conflicting logical systems, so why did the one we use today develop. It was because it best suited the world we lived in, if a human thought illogically, they died. Thus i think our logic and in turn mathematics is eternal truth, as it was developed from the workings of the natural world, if it was anything else we would not be here. Evolution shaped our logic to reflect the natural worlds most basic truths, and that is why we should put our faith in mathematics.

 

[lostcauses10x]

so why is physics not just a subset of mathematics?

Objects can and do exist in math and the relations there of.

Such may or may not exist in the physical world and its measurable (qualifier of quantity) relations. To the mathematician it should not mater if it is in the physical world.

Of course the physicist does if it is an observable and measurable relation.

A smart scientist will of course observe such in math. Such may be useful in new discovery, as so many times happens in mathematics and science.

 

[pwsnafu]

The interesting thing is, why only one logic? We cannot have two conflicting logical systems, so why did the one we use today develop. It was because it best suited the world we lived in, if a human thought illogically, they died. Thus i think our logic and in turn mathematics is eternal truth, as it was developed from the workings of the natural world, if it was anything else we would not be here. Evolution shaped our logic to reflect the natural worlds most basic truths, and that is why we should put our faith in mathematics.

Humans seem to have no problems rejecting http://en.wikipedia.org/wiki/Principle_of_bivalence" paradoxes), but math as we know it requires it.

 

[Functor97]

Humans seem to have no problems rejecting http://en.wikipedia.org/wiki/Principle_of_bivalence" paradoxes), but math as we know it requires it.

Please extend on this claim as i am not sure what you are claiming exactly.
If humans have trouble believing or understanding something, as it goes against their intuition that does not mean that it is wrong.
The nature of these paradoxes is interesting and i do not pretend to have an accurate answer, but the existence of a paradox does not mean mathematics is not based upon logic, it simply means our logic cannot explain said paradox. I did not claim our logical basis was perfect, but it is the best approximation of truth we can aim for.

 

[Studiot]

Strangely enough so far, mankind is the only thing that seems to use mathematics in our world, and so far the universe; as a form of communication

I don't think that biologists would agree with you.

I am no expert in biology but I do believe various creatures have been shown to possesses and use the ability to count.

Further look up ' the waggle dance' performed by honey bees.

http://www.google.co.uk/#hl=en&suge...gc.r_pw.&fp=35380f15a1864cdf&biw=1024&bih=585

 

[Functor97]

I don't think that biologists would agree with you.

I am no expert in biology but I do believe various creatures have been shown to possesses and use the ability to count.

Further look up ' the waggle dance' performed by honey bees.

http://www.google.co.uk/#hl=en&suge...gc.r_pw.&fp=35380f15a1864cdf&biw=1024&bih=585

but is "counting" mathematics, how do the animals percieve space, change, order and value?

 

[Studiot]

Like i said, i think all thought is mathematical, based upon logical axioms.

I know you said it, but I also thought we had agreed that there are some processes, such as the ones I exhibited, which are not based on mathematical thought.

Without such agreeement of terminology the discussion reduces from one of substance to one of semantics.

 

[Studiot]

how do the animals percieve space, change, order and value?

Actually all those are represented in the waggle dance!

 

[Functor97]

I know you said it, but I also thought we had agreed that there are some processes, such as the ones I exhibited, which are not based on mathematical thought.

Without such agreeement of terminology the discussion reduces from one of substance to one of semantics.

I do not think those activies are void of mathematical thought. I think they are so "diluted" and unrigorous that it appears to be the case that they are non mathematical, but physics appeard that way to the followers of Aristotle...

 

[lostcauses10x]

I don't think that biologists would agree with you.

I am no expert in biology but I do believe various creatures have been shown to possesses and use the ability to count.

Further look up ' the waggle dance' performed by honey bees.

http://www.google.co.uk/#hl=en&suge...gc.r_pw.&fp=35380f15a1864cdf&biw=1024&bih=585

You will also find a great controversy over this stuff. A lot say it is BS. It may be just a dance to say foraging should be done in general.

I will say they do warn intruders with a dance. Them African mixed ones will warn with a dance further away from there hive also. Personal experience.
A friends hobby is bees, including capturing swarms in residential areas. Learned a lot from him and direct observations. Interesting creatures.

 

[pwsnafu]

Please extend on this claim as i am not sure what you are claiming exactly.
If humans have trouble believing or understanding something, as it goes against their intuition that does not mean that it is wrong.
The nature of these paradoxes is interesting and i do not pretend to have an accurate answer, but the existence of a paradox does not mean mathematics is not based upon logic, it simply means our logic cannot explain said paradox. I did not claim our logical basis was perfect, but it is the best approximation of truth we can aim for.

Wait a sec, aren't you claiming that mathematical logic is the basis of human thought? If so, then mathematical logic uses bivalence, and therefore cannot be the basis of human thought which has no problems rejecting it.

You said "only one logic" in a previous post. Rejection of bivalence creates other forms of logic. Humans are able to go from one form to another without problems.

 

[Functor97]

Wait a sec, aren't you claiming that mathematical logic is the basis of human thought? If so, then mathematical logic uses bivalence, and therefore cannot be the basis of human thought which has no problems rejecting it.

You said "only one logic" in a previous post. Rejection of bivalence creates other forms of logic. Humans are able to go from one form to another without problems.

Intuitionist mathematicians reject bivalence and still seem to do work in mathematics. Read about Brouwer and the constructivists.

 

[pwsnafu]

Intuitionist mathematicians reject bivalence and still seem to do work in mathematics. Read about Brouwer and the constructivists.

So you proved my point. For example intuitionistic arithmetic has some very different results to that of Peano. You and I can move from one to the other without problem. Mathematical truths however do not.

 

[Functor97]

So you proved my point. For example intuitionistic arithmetic has some very different results to that of Peano. You and I can move from one to the other without problem. Mathematical truths however do not.

I think you have misunderstood my point, you just claimed mathematics could not be done without bivalence and i explained that it can, i think your ideas about a priori mathematical truths are wrong. I believe we choose the logical foundation and then apply that foundation, that application is in essence mathematics. We can choose different fundamental starting points, but the application of those axioms will be mathematics. That is my point, as our logical axioms are our most basic tautologies, the entire cerebral realm is built around them, thus our physics which at first appears quite physical, is actually just applied mathematics. The initial composition of logical tautologies is arbitary, but the application is always mathematical.
I really think you need to grasp the fact that intuitionistic mathematics is still mathematics.

 

[pwsnafu]

That is my point, as our logical axioms are our most basic tautologies, the entire cerebral realm is built around them, thus our physics which at first appears quite physical, is actually just applied mathematics.

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 really think you need to grasp the fact that intuitionistic mathematics is still mathematics.

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.

 

[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?