Mathematics, Discovered or Invented?

[Ostronomos]

I find it interesting where people will draw the line between discovered math and invented math. For you, it is calculus. For @sayak83 it seems to be partial differential equations (which are used extensively in quantum mechanics).

I see both as extensions of language that we use to model things we are interested in. But I have very serious doubts about the 'existence' of these systems outside of our minds.

I thought this would be an opportune time to point out that your serious doubts are due to the false belief in materialism. The fact that you are reading these words is proof that language is an object of a very different kind to matter. Welcome to the matrix, Neo. If we open our hearts and minds beyond the world of space and time then we may begin to see what is possible.

For example, you mention the natural numbers as 'God given' (following Kronecker). But take a look at Graham's number: Graham's number - Wikipedia

If we suppose that the observable universe is one trillionth of the 'real universe' and that ours in only one of 10^10^10^10 possible universes (far more than string theory postulates), and if we imagine each of these universes lasting 100 trillion years, and if we count the number of Plank units (plank volume times Plank time) for each of these universe, then the number of possible rearrangements of all of these Plank units throughout all of the possible universes is far, far, far smaller than Graham's number.

Fascinating.

Given that, in what possible sense does Graham's number 'actually exist'?

As far as I can see, it *only* exists in our minds as a construct in a formal system. anything outside of that is excluded by its size.

Our mind IS reality in mental form. You are creating a false dichotomy between mind and reality. I and a few other cosmologist have pointed this out numerous times.

Mind = Reality = Language. Once understood, this is strikingly obvious to even a casual observer.

My conclusion is that even the natural numbers cannot ALL exist. Only a very small part of that collection can 'really exist'.

This argument is flawed for a number of reasons because even the imaginary numbers exist as a mathematical language in the mind.

For those of you who may not have made the connection; existence can be placed on paper as language.

The same can be said for the real numbers, differential equations, etc.

So, in answer to @vulcanlogician and @sayak83, I think we *invent* formal systems to mimic what we see. We are then amazed that those formal systems continue to mimic what we see. But those actual formal systems go far, far beyond what is testable. They have 'objects' that are simply too large to actually exist in any conception of a multiverse. But these systems are extensions of our ability to have language. We *invent* the ways to describe things. We choose systems that are flexible in their ability to describe. And that vast, vast majority of what those formal systems have has *nothing* to do with anything physical even in the remotest extension we can imagine.

So, no, I am not surprised that occasionally our patterns of thought show up in how we look at the universe around us.

Even here you are hinting at a mathematical reality, yet fail to see the obvious.

 

[Polymath257]

I thought this would be an opportune time to point out that your serious doubts are due to the false belief in materialism. The fact that you are reading these words is proof that language is an object of a very different kind to matter. Welcome to the matrix, Neo. If we open our hearts and minds beyond the world of space and time then we may begin to see what is possible.


Fascinating.

Our mind IS reality in mental form. You are creating a false dichotomy between mind and reality. I and a few other cosmologist have pointed this out numerous times.

Mind = Reality = Language. Once understood, this is strikingly obvious to even a casual observer.

This argument is flawed for a number of reasons because even the imaginary numbers exist as a mathematical language in the mind.

For those of you who may not have made the connection; existence can be placed on paper as language.

Even here you are hinting at a mathematical reality, yet fail to see the obvious.

Yes, mathematics is, largely, a type of language. As such, it exists only within minds. it has no external objective existence. That is my whole point.

As such, the basic rules of math are invented and the consequences of those rules are discovered. At least, that is my conclusion after years of study and thinking about these issues. I might suggest that what you see as 'obvious' might, in fact, be false. This has happened many times in the study of math itself, so why not in more generality?

For example, if you go back 200 years, it was 'obvious' that no curve could fill an entire area or an entire volume. And yet, we now know that this is possible and we can write down specific curves that do so.

Usually, what is obvious is a matter of conditioning and thereby culture. Because of that, obviousness is a very poor indicator of truth. Instead, we need to *test* even the most obvious ideas to see if they actually hold in practice.

 

[Heyo]

I disagree. Ideals only exist in the mind, from all I can see.

You said in your first contribution that you would "go with a mixture of discovery and invention".
I think that the discovered parts (whichever that is for you) are the ideals which exist outside the mind.

 

[Polymath257]

You said in your first contribution that you would "go with a mixture of discovery and invention".
I think that the discovered parts (whichever that is for you) are the ideals which exist outside the mind.

But the discovered parts are the consequences of the rules that are invented.

Again, chess is invented, but we can discover the solution to a chess problem.

 

[Brickjectivity]

One might say that we discovered one of the many sets of rules, selecting them for our purpose and named them chess. Another set of rules we named checkers, another Othello. Naming and claiming is different from making them possible. They were always possible, hence always out there to be discovered. The rules exclude certain situations. How is that different from the rules by which our universe exists?

 

[Ostronomos]

Yes, mathematics is, largely, a type of language. As such, it exists only within minds. it has no external objective existence. That is my whole point.

Yet I have demonstrated that it does have objective existence. Based solely on the fact that it exists in the external world.

Have you not read the paper Quantum Metamechanics by Christopher Langan? Both he and I posit that anything that can be identified exists in some form or another. Including thoughts. Therefore they can interact with the external world. Which brings me to my next observation:

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

Please don't return to the forums deprived of all logic yet again.

 

[sayak83]

No, it says something about right triangles *if Euclidean geometry is assumed*. But, since the early 1800's, we know that there are non-Euclidean geometries that are just as internally consistent as Euclidean geometry. In these geometries, such basic 'facts' as Pythagorus' theorem and that the sum of the angles of a triangle is a straight angle are simply false. Furthermore, such geometries are more appropriate in many situations (for example, spherical geometry is non-Euclidean as is the geometry of general relativity).

I can easily imagine situations where an alien race would first arrive at a non-Euclidean geometry where Pythagorus' theorem fails.

Unlike mathematics, the motions of the bodies is objectively measurable. As long as those motions are correctly predicted, we have a *physically* equivalent theory.

This has a long history even in human physics. Newton described gravity in terms of a force which acts at a distance. Lagrange described it as a 'potential' that acts locally. There is also description where the motion maximizes an 'action' in a way that has been interpreted as teleological. These theories are *completely* equivalent in the motions they predict and are considered by physicists as just different ways to approach certain physics problems. That they are philosophically very different is simply seen as irrelevant.

Einstein described it as a distortion of the geometry of spacetime. But there is also a Lagrangian description of Einsteinian gravity. Again, these are equivalent in the predictions of motion, which is ALL that is important for the physics.

So, no, I would NOT necessarily expect that an alien race would arrive at the same *laws* of physics as we have (even to the current approximation), but I *do* expect the two systems to make the same predictions about observable motion.

I see it as easily conceivable. if their planet is small enough, or the history of their math is different enough, they may easily have come to spherical geometry as the basis of their geometrical thinking. And in that system Pythagorus' theorem is simply false.

Let's look at a very simple case. All motions have a velocity. And velocity is V= dX/dt. This is a mathematical relationship that is related three physical observables : position x, velocity v and time t. (In QM there will be a different relationship here, but let's go with that). Now we have three physical observables that really exist as observables x, v, t and a relationship that observably exist between the three as well.
Now what is the ontological status of this relation in your view? Does the relationship really exist independent of minds, or it is also a creation of the mind?
I will go to the next question after you answer this for me.

 

[Polymath257]

Let's look at a very simple case. All motions have a velocity. And velocity is V= dX/dt. This is a mathematical relationship that is related three physical observables : position x, velocity v and time t. (In QM there will be a different relationship here, but let's go with that). Now we have three physical observables that really exist as observables x, v, t and a relationship that observably exist between the three as well.
Now what is the ontological status of this relation in your view? Does the relationship really exist independent of minds, or it is also a creation of the mind?
I will go to the next question after you answer this for me.

It is a creation of our minds. We choose how to measure distance and time and to construct the ideas of velocity.

If you go back to Aristotle, the concept of velocity as a ratio of distance to time would have been unintelligible. At that point, only ratios of like quantities would have been accepted as meaningful.

 

[Polymath257]

Yet I have demonstrated that it does have objective existence. Based solely on the fact that it exists in the external world.

Really? Where was that shown? Where does it exist in the external world?

Have you not read the paper Quantum Metamechanics by Christopher Langan? Both he and I posit that anything that can be identified exists in some form or another. Including thoughts. Therefore they can interact with the external world. Which brings me to my next observation:

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

Please don't return to the forums deprived of all logic yet again.

And you call Langan’s material logic? Sorry, but I don’t find it reasonable, logical, or even more than drivel. As the “smartest person in the world”, he seems remarkably stupid about conspiracy theories.

Thoughts exist as patterns in our brains. But that doesn’t mean content of the thoughts corresponds to real things.

 

[muhammad_isa]

Thoughts exist as patterns in our brains. But that doesn’t mean content of the thoughts corresponds to real things.

Hmm .. let's get this straight..
Are you saying that 'number' is not a real thing?

.. 'number' as in counting a number of objects.

 

[Polymath257]

Hmm .. let's get this straight..
Are you saying that 'number' is not a real thing?

.. 'number' as in counting a number of objects.

Numbers only exist in our minds, not outside of them. They are real only in the sense that they are useful for helping us understand reality. They are abstractions, not real entities.

For small integers, this isn’t as clear as it is, for example, with Grahams number (see above).

 

[muhammad_isa]

..They are abstractions, not real entities.

For small integers, this isn’t as clear as it is, for example, with Grahams number (see above).

No, it's not clear at all to me.
I just don't see how the difference between 1 and 2, for example, is merely an "abstraction" ..
..and we can all agree on that "unreal" abstraction.

 

[sayak83]

It is a creation of our minds. We choose how to measure distance and time and to construct the ideas of velocity.

If you go back to Aristotle, the concept of velocity as a ratio of distance to time would have been unintelligible. At that point, only ratios of like quantities would have been accepted as meaningful.

If velocity is mind created, a moving object has no actual property called the velocity. Perhaps one could say the same with regards to position, time etc as well?
Then, I am curious, what is the ontology of the mind independent reality according to you?

 

[blü 2]

I think @Heyo was getting at the fact that it is common in math for the same idea to appear independently in different places and even at different times.

One example is the development of calculus, which is attributed to both Newton and Leibnitz (although others did contribute). Both came up with essentially the same techniques independently.

The 'where' of discovery seems to be of less interest, though, than the 'who' and 'when'. Of course, different concepts in math arose at different times. Even today it is common for two researchers to find essentially the same results and techniques even though they have no contact.

That does suggest 'discovery'.

Indeed. I recall reading that Riemann formulated and proved assertions about integrals in algebraic functions, and these were later found to have been formulated and proved in a letter by Galois (on the eve of his famous fatal duel) twenty years earlier.

It seems reasonable to say that mathematical new ideas are enabled by the progress of mathematical ideas, which give the attuned mind a place to stand to make further new perceptions ─ I recall reading of a 19th century Eastern European or Russian mathematician (Bolyai?) reporting his ideas in a letter to his father, and his father telling him to publish forthwith because obviously the tide is coming in for such perceptions but first to publish gets all the glory.

 

[Heyo]

But the discovered parts are the consequences of the rules that are invented.

Numbers only exist in our minds, not outside of them. They are real only in the sense that they are useful for helping us understand reality. They are abstractions, not real entities.

So, not the faintest bit of idealism in you? I thought I could reconcile materialism with idealism by treating them as (mostly) non-overlapping magisteria. But I'm out of arguments, and I guess, so are you. One has to believe in ideals - or not, there is no rational way to or away from it.
We'll have to agree to disagree.

 

[☆Dreamwind☆]

Invented

 

[Polymath257]

If velocity is mind created, a moving object has no actual property called the velocity. Perhaps one could say the same with regards to position, time etc as well?

Will, at the very least, all of those depends on the frame used, so are not properties of the object, but properties of the description.

Then, I am curious, what is the ontology of the mind independent reality according to you?

I haven’t stabilized on one. One big consideration is that realism: that things have definite properties, is problematic. As far as I can see, we have ways of interacting. And that is really all we have.

 

[Polymath257]

So, not the faintest bit of idealism in you? I thought I could reconcile materialism with idealism by treating them as (mostly) non-overlapping magisteria. But I'm out of arguments, and I guess, so are you. One has to believe in ideals - or not, there is no rational way to or away from it.
We'll have to agree to disagree.

In my view, universals are mental. So I guess I am close to being a minimalist as oppressed to a realist on that matter.

In general, realism has issues with modern physics.

 

[Ostronomos]

*shakes head*

"Materialism... the belief of fools".

 

[Ostronomos]

Let me spell it out for you in crayons...

The mind exists in reality as information, and is therefore as much a part of reality as matter.