Is there a materialist explanation of mathematics?

[GaryTaylor]

The concepts “true” and “provable” are different. If we are talking about statements in number theory, then the concept of “true” has been assigned meaning prior to the construction of any particular system of axioms. Whether or not a given statement is provable depends on the particular system. If the system of axioms is inconsistent, then all statements are provable, but it does not follow that all statements are in fact both true and false. If the system of axioms is incomplete, then at least one true statement is not provable from those axioms.

I would say:
A statement is a theorem if it is provable from the axioms. Truth in number theory is a topic that goes beyond any particular system of axioms.

How so…

then the concept of “true” has been assigned meaning prior to the construction of any particular system of axioms.

or

Truth in number theory is a topic that goes beyond any particular system of axioms

 

[inocente]

It is not a quibble. It is one of the sources of our disagreement with each other.

I disagree. The concepts “true” and “provable” are different. If we are talking about statements in number theory, then the concept of “true” has been assigned meaning prior to the construction of any particular system of axioms. Whether or not a given statement is provable depends on the particular system. If the system of axioms is inconsistent, then all statements are provable, but it does not follow that all statements are in fact both true and false. If the system of axioms is incomplete, then at least one true statement is not provable from those axioms.

Thus, rewriting your claim, I would say:
A statement is a theorem if it is provable from the axioms. Truth in number theory is a topic that goes beyond any particular system of axioms.

That last sentence sounds like a religious statement.

I’d just point out that “true” means “in accordance with fact or reality”, and we need to take note of that “or” - not all facts are part of reality.

For instance you can imagine a world where gravity works the opposite way to our world, and you can list true facts about the consequences in that world, but none of those facts would be true in our reality, in our world.

If your last sentence is true then number theory must necessarily be true in all possible worlds. Can you prove that?

 

[GaryTaylor]

So much better, and all the math students you scared off just returned.

 

[PseuTonym]

I’d just point out that

Is this a matter of popular usage of the words “true”, “fact”, and “reality” and your intuitions about language? Or is this based on your reading of some philosophical writings or reference sources that stipulated somewhat specialized meanings for those words?

I’d just point out that “true” means “in accordance with fact or reality”, and we need to take note of that “or” - not all facts are part of reality.

When you write “not all facts are part of reality”, you are making a claim that is unclear to me because, for one thing, I do not know what you mean in that context when you use the word “reality.”

Earlier in this thread, there was at least one use of the word “reality” where it seemed that the person who was using it meant “material reality.”

Here is the how the word “reality” was used there:
“I think the mathematical models we build of reality are approximations and their isomprphic qualities are good within ranges.”
Here is a link to the post:
forums.catholic-questions.org/showpost.php?p=12760595&postcount=39

Here is a link to my reply:
forums.catholic-questions.org/showpost.php?p=12761113&postcount=45

For instance you can imagine a world where gravity works the opposite way to our world, and you can list true facts about the consequences in that world, but none of those facts would be true in our reality, in our world.

“True facts” is a strange combination of words. Provided that we are dealing with statements that are clear enough, we can consider the possibility that one such statement is true. We can also consider the possibility that one such statement is false. The truth of a statement is different from how we come to know or believe that the statement is true. The falsehood of a statement is different from how we come to know or believe that the statement is false.

 

[PseuTonym]

Oh, I see what you mean. Since the usual, but not always, independent variable in differential calculus is time, the question is whether or not time is discrete.
I am not sure whether or not space or time is discrete. It is possible, I suppose. **In any case, the real line is not discrete in a mathematical sense, since between any two points on the real line, you can find one in between. **This would not be true AFAIK with a thin rod.

Your thought process is somewhat mysterious.

Originally, you wrote:
“The differential calculus is built on the assumption of the continuity of the real line, and yet AFAIK, matter is discrete and not continuous. therefore the math model is in this case an approximation.”

More recently, you wrote (see above for the full quote):
“In any case, the real line is not discrete in a mathematical sense, since between any two points on the real line, you can find one in between.”

Given any two rational numbers, there is a rational number between them. So it’s becoming less clear (rather than more clear) what you meant originally when you wrote about “the assumption of the continuity of the real line.”

Do you know of a way to develop differential calculus using merely the set of rational numbers rather than the set of real numbers? That is not a rhetorical question. There is a mathematician named “Norman Wilberger” (who has videos on Youtube and also his own website) and he would probably be very interested in such a development.

Now, relating back to the topic of this thread,

Mathematical representations of space and time can be very good approximations, but they are not exact. Or perhaps I should say that eventually there might be a mathematical representation of space and time that is exact, but we would not know that it is exact. There are at least two reasons for this. First, our measurements have limited precision. At best, we could say the correct value of something lies within some interval, but it is an interval. If the interval is made too small then we can no longer say whether or not the correct value lies in the interval. There are also sources of error. So if we want to reduce the chance of making a statement that will eventually be discovered to be definitely false, then we need a safety margin that makes our interval bigger, rather than smaller.

That is why, if we are interested in focusing on mathematics that may be difficult to explain from the point of view of materialist philosophy, then we should focus on mathematics that deals with the properties of non-negative integers.

 

[GaryTaylor]

Right so we are talking scientific law and theory and the interaction.

As such, a law is limited in applicability to circumstances resembling those already observed, and may be found false when extrapolated.

 

[Tomdstone]

Your thought process is somewhat mysterious.

Originally, you wrote:
“The differential calculus is built on the assumption of the continuity of the real line, and yet AFAIK, matter is discrete and not continuous. therefore the math model is in this case an approximation.”

More recently, you wrote (see above for the full quote):
“In any case, the real line is not discrete in a mathematical sense, since between any two points on the real line, you can find one in between.”

Given any two rational numbers, there is a rational number between them. So it’s becoming less clear (rather than more clear) what you meant originally when you wrote about “the assumption of the continuity of the real line.”

Do you know of a way to develop differential calculus using merely the set of rational numbers rather than the set of real numbers? That is not a rhetorical question. There is a mathematician named “Norman Wilberger” (who has videos on Youtube and also his own website) and he would probably be very interested in such a development.

Now, relating back to the topic of this thread,

Mathematical representations of space and time can be very good approximations, but they are not exact. Or perhaps I should say that eventually there might be a mathematical representation of space and time that is exact, but we would not know that it is exact. There are at least two reasons for this. First, our measurements have limited precision. At best, we could say the correct value of something lies within some interval, but it is an interval. If the interval is made too small then we can no longer say whether or not the correct value lies in the interval. There are also sources of error. So if we want to reduce the chance of making a statement that will eventually be discovered to be definitely false, then we need a safety margin that makes our interval bigger, rather than smaller.
.

By the term “real line” is meant “mathematical real line”.

 

[Tomdstone]

That is why, if we are interested in focusing on mathematics that may be difficult to explain from the point of view of materialist philosophy, then we should focus on mathematics that deals with the properties of non-negative integers.

By doing so, you will be omitting a lot of mathematical developments.

 

[GaryTaylor]

By the term “real line” is meant “mathematical real line”.

 

[inocente]

Is this a matter of popular usage of the words “true”, “fact”, and “reality” and your intuitions about language? Or is this based on your reading of some philosophical writings or reference sources that stipulated somewhat specialized meanings for those words?

Google “true definition”. It’s also the first definition given in the Oxford - “In accordance with fact or reality”.

When you write “not all facts are part of reality”, you are making a claim that is unclear to me because, for one thing, I do not know what you mean in that context when you use the word “reality.”

I gave the example of reverse gravity in a different world. You can infer facts about that world but they are not facts about this world. Since we have the Oxford open, look up “fact” - it’s a “a thing that is indisputably the case”. They are indisputably the case in the other world but not in ours.

Surely none of this is controversial?

*Earlier in this thread, there was at least one use of the word “reality” where it seemed that the person who was using it meant “material reality.”

Here is the how the word “reality” was used there:
“I think the mathematical models we build of reality are approximations and their isomprphic qualities are good within ranges.”
Here is a link to the post:
forums.catholic-questions.org/showpost.php?p=12760595&postcount=39*

Here is a link to my reply:
forums.catholic-questions.org/showpost.php?p=12761113&postcount=45

Take something like E=mc[sup]2[/sup]. We have good reason to believe it’s always a true representation of reality, but we can’t prove it because it’s always possible that someone may find a counter-example tomorrow.

So if we can’t even prove that mathematical models are true in the material world, how could we possibly do it for “non-material reality”?

“True facts” is a strange combination of words. Provided that we are dealing with statements that are clear enough, we can consider the possibility that one such statement is true. We can also consider the possibility that one such statement is false. The truth of a statement is different from how we come to know or believe that the statement is true. The falsehood of a statement is different from how we come to know or believe that the statement is false.

I don’t think you’d say that if you were a cop who found a bag of money from a bank raid in my car, and I told you someone must have put it there when I wasn’t looking, honest officer. You would want a chain of evidence before accepting the truth of my statement.

 

[GaryTaylor]

Take something like E=mc2. We have good reason to believe it’s always a true representation of reality,

Right, thus represented by various theories past the known of verification.

askamathematician.com/2011/03/q-why-does-emc2/

Side note: This derivation isn’t a “proof” per say, just a really solid argument: “there’s this thing that’s conserved according to relativity, and it looks exactly like energy”. However, you can’t, using math alone, prove anything about the outside universe. The “proof” came when E=mc2 was tested experimentally (with particle accelerators ‘n stuff).

But Einstein’s techniques and equations have been verified as many times as they’ve been tested. One of the most profound conclusions is that, literally, “energy is the time component of momentum”. Or “E/c” is at least. So conservation of energy, momentum, and matter are all actually the same conservation law!

 

[PseuTonym]

By the term “real line” is meant “mathematical real line”.

Of course, the word “real” in the context of “real line” is a technical term of mathematics, and is not an adjective form of the noun “reality.” However, ordinarily people mix up the idea of the real line with some very specific assumptions about the kind of set theory that is being used, and attribute conclusions that might depend upon the underlying set theory to the real line.

There is also an opposite tendency to claim that the real numbers are a complete ordered field, and to claim that any property of real numbers (deduced from any particular set-theoretic construction of the set of real numbers) that go beyond the list of properties of a complete ordered field is an unintended consequence of the particular construction, and has no mathematical significance. However, we have the opposite situation for the non-negative integers, with Goedel’s incompleteness theorem producing an acknowledgement that no recursively enumerable system of axioms gives a complete description of the properties the non-negative integers.

By the term “real line” is meant “mathematical real line”.

That answers a question that I did not ask. I was seeking clarification of what you had in mind when you were applying the adjectives “discrete” and “continuous.” You started with “continuous”, and then jumped to “discrete”, and when you jumped back to “continuous” it seemed that the word “continuous” lost some of its meaning, depending upon what you meant by it in the first place.

I will ask again a question already asked …

Do you know of a way to develop differential calculus using merely the set of rational numbers rather than the set of real numbers?

 

[PseuTonym]

PseuTonym:
“That is why, if we are interested in focusing on mathematics that may be difficult to explain from the point of view of materialist philosophy, then we should focus on mathematics that deals with the properties of non-negative integers.”

By doing so, you will be omitting a lot of mathematical developments.

There are three things to consider: what you mean, why you think it matters, and why you think it is true.

Let us begin with what you mean. The word “developments” is a bit vague. On the one hand, the various branches of mathematics are interconnected. We cannot ignore any branch of mathematics, because any branch of mathematics might produce theorems that provide insight that allows for new conjectures in number theory to be formulated, and for old conjectures in number theory to be resolved.

On the other hand, there might be a conjecture in number theory that is true but unprovable, and that will remain unprovable until such time as new axioms are formulated, explained, and believed. However, the thing about belief is that it does not guarantee that what is believed is actually true. Mathematicians are very hesitant about accepting new axioms because of the risk that a contradiction will eventually be deduced. So for long periods of time, change occurs inside the box, while the box itself stays the same.

Thus, there are two kinds of developments in mathematics: those inside a fixed box, and those that change the box.

From my point of view, a focus on the non-negative integers does not imply that we ignore developments inside the box, and it does not imply that we ignore developments that change the box. However, let us suppose that I am wrong about that. What difference would it make?

This thread poses the question: “can materialist philosophy provide an explanation of mathematics?” The answer will be “No” if there is one branch of mathematics (specifically, number theory) that materialist philosophy cannot explain. Even if you convinced yourself that materialist philosophy could explain all other branches of mathematics, the answer would still be “No.”

 

[Tomdstone]

This thread poses the question: “can materialist philosophy provide an explanation of mathematics?” The answer will be “No” if there is one branch of mathematics (specifically, number theory) that materialist philosophy cannot explain. Even if you convinced yourself that materialist philosophy could explain all other branches of mathematics, the answer would still be “No.”

True. Have you shown how number theory disproves a materialist philosophy of mathematics?

 

[Tomdstone]

Do you know of a way to develop differential calculus using merely the set of rational numbers rather than the set of real numbers?

An attempted calculus using the set of rational numbers only will be incomplete since the limit of a sequence of rationals converging to pi will not exist in the set of rationals.

 

[tonyrey]

What is the reason that they are self-evident? Is it because they refer to facts about material reality - like numbers?

Disagreement does not imply that nothing is self-evident nor do the symbols we use imply that the aspects of reality - like the molecular structure of water - described by numbers don’t exist.

By chance?

For instance, a great deal of math has been developed for string theory, but string theory may well fail when it is tested and so be completely wrong. None of that math will then have been found to be part of reality.

What about that which corresponds to reality? Is that a coincidence?

 

[inocente]

Disagreement does not imply that nothing is self-evident nor do the symbols we use imply that the aspects of reality - like the molecular structure of water - described by numbers don’t exist.

Agreed. confused

What about that which corresponds to reality? Is that a coincidence?

No. As I think I said earlier, the causal connection is that anything which is orderly can be described by math; our world contains some order; and therefore math can be used to describe that order (such as in the physical law). But of course it doesn’t then follow that all math necessarily describes order in our world.

 

[tonyrey]

Agreed. confused

No. As I think I said earlier, the causal connection is that anything which is orderly can be described by math; our world contains some order; and therefore math can be used to describe that order (such as in the physical law). But of course it doesn’t then follow that all math necessarily describes order in our world.

I agree. That is why no materialist explanation of mathematics is possible. Numbers are intangible and presuppose insight.