Mathematics, Discovered or Invented?

[Polymath257]

This all started with you saying "the ideas in the mind don’t necessarily code for reality."

Exactly. Out ideas can be wrong. We can be mistaken. What we think may not be correct. In fact, it is possible to have full confidence in one's ideas and to still be factually mistaken.

The question then becomes how we avoid such error. And, the best way we have found is to test our ideas: we *try to prove them wrong* and see which ideas survive such attempts. Only those that survive many repeated attempts to disprove them can be said to have some facet of knowledge (justified true belief).

It is because we can be wrong, even in our most cherished beliefs, that we need to be skeptical and avoid taking *anything* on faith alone.

..but math is not an invention of mankind .. it is discovered .. it existed in the same way 1000 years ago as it does today.

Once again, there is a place for discovery. Once the rules have been *invented* (to adapt to our intuitions), we *discover* the consequences of those rules. Just like in the game of chess: we invent the rules and discover their effects.

 

[muhammad_isa]

We can be mistaken. What we think may not be correct..

That is obvious, I would have thought..

..the best way we have found is to test our ideas: we *try to prove them wrong* and see which ideas survive such attempts..

Yes .. but that is NOT confined to physical experiment.
It includes logic and many other disciplines.

It is because we can be wrong, even in our most cherished beliefs, that we need to be skeptical and avoid taking *anything* on faith alone..

Well, I agree .. "faith" as in believing something without qualification, is not reasonable.

Once again, there is a place for discovery. Once the rules have been *invented* (to adapt to our intuitions), we *discover* the consequences of those rules..

Mathematics is not like that .. it is a cosmic reality, and not one that only exists in a few people's minds.

 

[Polymath257]

Mathematics is not like that .. it is a cosmic reality, and not one that only exists in a few people's minds.

I disagree. As I see it, math is a language we have invented (as is logic, by the way). It exists in our minds (and, in a very limited way, in the minds of other animals).

But let's get specific. If you think the number '2' exists independent of any minds, exactly what is it? Does it exist as many individuals, one for each system of numbers? Is there a different '2' for every possible model of Peano axioms? How is it that the '2' for two apples is the same as the '2' for two oranges?

 

[muhammad_isa]

How is it that the '2' for two apples is the same as the '2' for two oranges?

I think you confuse notation (or language) with the actual concepts.
The concept of one or two is known by all of us .. it's universal.

..and I see no merit in exercising my skills in English to explain the difference between apples and oranges.

 

[Polymath257]

I think you confuse notation (or language) with the actual concepts.
The concept of one or two is known by all of us .. it's universal.

..and I see no merit in exercising my skills in English to explain the difference between apples and oranges.

And what is the “actual concept”? Be specific.

I am not talking about notation. I am talking about the fact that there are different objects that are ‘2’ in different mathematical systems. All have a certain relationship to the overall system while being different in specifics.

For example, in Von Neumann’s system, 2 is the set {{},{{}}}. In Zormelo’s system, 2 is the set {{{}}}. For the integers, 2 is an equivalence class of pairs of natural numbers of the form (n++,n). In the system of real numbers, one definition would have 2 be a subset of the collection of rational numbers and another definition would have it be a certain equivalence class of Cauchy sequences of rational numbers. None of these are the same as any of the others.

This is why I ask if there are several different things that are all ‘2’. Is’2’ as a positive integer that same as ‘2’ as a rational number and ‘2’ as a real number? Because the specifics in each case are very different.

What, precisely, is the number ‘2’? More generally, what precisely is a ‘number’?

It might be a good idea to read Benaceraf’s paper concerning what numbers cannot be.

I would also point out that it is possible to have a mathematical system that allows talking about 2 objects without having an object that is the number 2. Being able to count does NOT imply that 2 exists as a separate entity.

 

[muhammad_isa]

And what is the “actual concept”? Be specific.

One does not need a PhD in Pure Maths to understand what the layman means by counting
2 apples or oranges.
Are you saying that you don't know what concept the layman refers to?

I am not talking about notation. I am talking about the fact that there are different objects that are ‘2’ in different mathematical systems. All have a certain relationship to the overall system while being different in specifics.

I am aware of that .. but why not just keep things simple for now?
i.e. the so-called "real" numbers

Of course, that's not to say that one cannot invent notations, such as Cartesian coordinates etc.
That is another issue.

For example, in Von Neumann’s system, 2 is the set {{},{{}}}. In Zormelo’s system, 2 is the set {{{}}}.

They are extensions/inventions, and not needed to understand the concept of number that we ALL
understand .. eg. children can understand

 

[Polymath257]

One does not need a PhD in Pure Maths to understand what the layman means by counting
2 apples or oranges.
Are you saying that you don't know what concept the layman refers to?

I understand the technique of counting. But no, I do not know what it means to be a number. I do not know what '2' is supposed to be *as an independently existing object*. I do know what it means *as an idea in our minds*. Do you see the difference?

I am aware of that .. but why not just keep things simple for now?
i.e. the so-called "real" numbers

You mean the ones with infinite decimal expansions? Those are the 'simple' ones?
Or are you actually meaning the natural numbers: 0, 1, 2, ...

Of course, that's not to say that one cannot invent notations, such as Cartesian coordinates etc.
That is another issue.

Agreed.

They are extensions/inventions, and not needed to understand the concept of number that we ALL
understand .. eg. children can understand

Those are attempts to actually define what is meant by the idea. it is much harder than you seem to understand.

And no, we understand the process of counting. I don't think we understand 2 as a separately existing entity outside of our minds.

 

[muhammad_isa]

..I don't think we understand 2 as a separately existing entity outside of our minds.

It's a concept, and a widely understood one.
You might not "believe" in non-material concept, but I for one, do.

 

[Polymath257]

It's a concept, and a widely understood one.
You might not "believe" in non-material concept, but I for one, do.

Concepts are in our minds, right? They don’t exist outside of our minds, right? That is all I am saying.

This is still true even if you believe in a nonmaterial mind. At least that’s how I see it.

 

[Ostronomos]

Exactly. Out ideas can be wrong. We can be mistaken. What we think may not be correct. In fact, it is possible to have full confidence in one's ideas and to still be factually mistaken.

This dichotomy only applies to the material or classical world. But what happens when that distinction dissolves? What happens when mind influences reality? It is said that consciousness is not located inside the skull, but is a field within which our brains and bodies exist. The double slit experiment can only work by this principle.

An axiom of metaphysics:

Reality contains mind while mind contains reality.

This is not to be mistaken as physical reality but rather, mental reality.

The question then becomes how we avoid such error. And, the best way we have found is to test our ideas: we *try to prove them wrong* and see which ideas survive such attempts. Only those that survive many repeated attempts to disprove them can be said to have some facet of knowledge (justified true belief).

Anyone who puts much faith in science at the exclusion of all else like theology is wrong. This is the height of intellectual vanity.

It is because we can be wrong, even in our most cherished beliefs, that we need to be skeptical and avoid taking *anything* on faith alone.

Agreed. As I have seen evidence for a God I hold dear the belief in God. You may not have seen evidence, but I have. In the form of universal consciousness. This is now part of modern science.

Once again, there is a place for discovery. Once the rules have been *invented* (to adapt to our intuitions), we *discover* the consequences of those rules. Just like in the game of chess: we invent the rules and discover their effects.

Mathematics on the other hand is discovered through observing nature.

 

[Ostronomos]

Concepts are in our minds, right? They don’t exist outside of our minds, right? That is all I am saying.

But the thing is they may exist outside of our minds.

Axiom of Metaphysics:

The higher dimension contains the separation, effecting the non-separation.

This is still true even if you believe in a nonmaterial mind. At least that’s how I see it.

You deserve credit for saying that that is your opinion. Thank you.

 

[muhammad_isa]

Concepts are in our minds, right? They don’t exist outside of our minds, right? That is all I am saying..

Well, I suppose it all depends on how you view the "cosmos" ..
When a concept exists in billions of minds, to think that it is not cosmically meaningful
is very dubious for me.

 

[Heyo]

Concepts are in our minds, right?

Yep.

They don’t exist outside of our minds, right?

Nope.

There are concepts, which we invented, like money or borders.
There are abstractions, like circles, which are stripped of all real properties and reduced to pure concepts.
And there are concepts that we can't see with our eyes, only with our minds. But they are not invented. Any sufficiently complex mind can "see" them, and two independent minds will agree that they "see" the same concept.

 

[Polymath257]

Well, I suppose it all depends on how you view the "cosmos" ..
When a concept exists in billions of minds, to think that it is not cosmically meaningful
is very dubious for me.

Cosmically meaningful? Meaningful to those people who understand it, possibly, but I really doubt that the cosmos cares about anything.

The cosmos is much, much, much larger than our small planet. A billion minds on a planet orbiting an ordinary star in one of billions of galaxies is NOT 'cosmically relevant'.

 

[Polymath257]

This dichotomy only applies to the material or classical world. But what happens when that distinction dissolves? What happens when mind influences reality? It is said that consciousness is not located inside the skull, but is a field within which our brains and bodies exist. The double slit experiment can only work by this principle.

And that is simply false. The double slit experiment has NOTHING to do with consciousness. It has to do with decoherence, which *any* sufficiently complex environment can manage.

An axiom of metaphysics:

Reality contains mind while mind contains reality.

I don't assume that axiom. The mind does NOT contain reality: it models it, imperfectly.

This is not to be mistaken as physical reality but rather, mental reality.

Which is a proper subset of physical reality.

Anyone who puts much faith in science at the exclusion of all else like theology is wrong. This is the height of intellectual vanity.

And I would say to to not go with the scientific method (test all ideas and require ideas to be testable) is to go beyond wisdom.

Agreed. As I have seen evidence for a God I hold dear the belief in God. You may not have seen evidence, but I have. In the form of universal consciousness. This is now part of modern science.

Mathematics on the other hand is discovered through observing nature.

 

[Polymath257]

Yep.

Nope.

There are concepts, which we invented, like money or borders.
There are abstractions, like circles, which are stripped of all real properties and reduced to pure concepts.
And there are concepts that we can't see with our eyes, only with our minds. But they are not invented. Any sufficiently complex mind can "see" them, and two independent minds will agree that they "see" the same concept.

Except that doesn't actually happen in practice. When asked to be more specific, people give wildly varying details. There are commonalities because of shared culture, but especially going across cultures we see a great deal of diversity.

 

[Polymath257]

But the thing is they may exist outside of our minds.

Axiom of Metaphysics:

The higher dimension contains the separation, effecting the non-separation.

You deserve credit for saying that that is your opinion. Thank you.

And I do not assume anything close to that axiom. Where did you find it? It seems like a bunch of woo to me.

 

[muhammad_isa]

Cosmically meaningful? Meaningful to those people who understand it, possibly, but I really doubt that the cosmos cares about anything.

When I say "cosmos", I refer to its known attributes (i.e. the physical universe), and also that which we
do NOT know.

..and that is where we "part company" .. because you refuse to work with the non-physical (i.e. the unknown)
You would rather presume that it doesn't exist, whereas I would not.

I see 'the study of mind' as more than an exploration of the physical.
I acknowledge that our thoughts can affect the physical brain, and vice-versa.
..whereas you presumably see the mind as a one-way phenomena .. or ignore that which can't be physically demonstrated.
Psychology is actually a science with a vast number of competing theories.

 

[Polymath257]

When I say "cosmos", I refer to its known attributes (i.e. the physical universe), and also that which we
do NOT know.

..and that is where we "part company" .. because you refuse to work with the non-physical (i.e. the unknown)
You would rather presume that it doesn't exist, whereas I would not.

I fully acknowledge there are many things we do not know. We don't know, for example, the masses of the neutrinos. We don't know the conditions for life to form. We don't know much at all about quantum gravity.

But, what I *do* want, before I consider something, is evidence it actually exists. otherwise it is way too easy to formulate infinitely many fantasies.

I see 'the study of mind' as more than an exploration of the physical.
I acknowledge that our thoughts can affect the physical brain, and vice-versa.
..whereas you presumably see the mind as a one-way phenomena .. or ignore that which can't be physically demonstrated.
Psychology is actually a science with a vast number of competing theories.

I'm not sure what you even mean by the mind being 'one way' or not.

As I see it, our minds are like the software running on the hardware of our brains. We collect information about the world through our senses and process that information, formulating a model of both the world and us in the world.

Software 'affects' the physical hardware (obviously) and the hardware affects the software (again, obviously). So yes, our thoughts can affect our bodies and vice versa. But the software cannot run without the hardware. Our minds only exist (as processes--running software) when there is the hardware to run them.

So, the study of computer software is different than the study of computer hardware. So yes, the study of the mind is more than the study of the brain. But just as we cannot have running software without the hardware, we cannot have a mind without a physical substrate (for us, the brain).

At least, that is how I see it.

 

[Polymath257]

Yep.

Nope.

There are concepts, which we invented, like money or borders.
There are abstractions, like circles, which are stripped of all real properties and reduced to pure concepts.

Circle only make sense if there is a notion of distance. For the circles you are thinking of, it probably requires a Euclidean plane. But we can define the concept in non-Euclidean contexts and beyond.

For example, if we use the 'taxicab' version of distance, circles are squares.