Applicability of Maths to the Universe

[Belacqua]

That said, let me share with you how I intuit numbers like 2. Based on how I currently see things, there is no number 2 floating out there in the Platonic sense and serving as some form of cause for the concept of 2 in our minds. For me, number 2 strictly exists in our minds, as a way to "visualize" a certain quantity of identical things. The quantity is out there in a "vague" sense, but it is not decipherable as "2" without a mind to see separateness and "identicalness" of the objects of interest. What would be the biggest challenge to this view?

This is all very difficult for me. As so often on this forum, I find myself in the role not of advocating a position but of wanting others to hold back from a position about which they may have too much confidence. I can't say any more than Popper, and his ideas have been worked out in far more detail than I'm aware of, I'm sure. 

One thing to ponder: numbers which no human has ever thought of yet. Imagine, for example, a very large prime number which no one has discovered. It has never appeared in the mind of any person. If numbers depend for their existence on appearing in minds, then this number doesn't exist. But even so, it may be more proper to say that it does exist, but hasn't been discovered yet. If it does exist, but has never been thought, then "where" is it?

But I put the "where" in scare quotes because of our bad habit of thinking of existence in spatial terms. It is begging the question if we say that anything which exists must exist in a location, with extension. This is our habit, especially in modern times, but I'm not sure it's always true. It would probably be better to ask HOW it exists, if it does so without location or extension. 

It may be that while numbers are the inventions of people, they exist not only as their appearance in individual minds. They are somehow commonly held, and exist even if no one is currently thinking them. 

The example Popper uses is about symphonies, which are also World Three objects in his system, like numbers. Imagine Beethoven's 5th symphony. The symphony itself is not identical with its score or its CD. Those are recordings made of it, but are not the symphony itself. It somehow continues to exist whether anyone is hearing it or not. 

I have used the example of Sherlock Holmes, a character everyone knows. Holmes was made up by a person's mind, but is no longer dependent on any individual mind. It may well be that at any given moment, no one in the world is imagining him. Yet he still somehow exists. And though he doesn't exist materially, and can't be the object of scientific studies, it is still possible to make statements about him which are correct or incorrect. And it is possible to distinguish between the "real" Sherlock Holmes ("real" in what way?) and a parody or "re-boot." 

And god knows that no one here wants to talk about theology, but all of this has been discussed for millennia. It is only recent and naive people who assume that the Christian God has a location, physical extension, material existence, etc. As a non-material thingy, it is not the subject of science, but is still said to exist in a way that material things do not.

[BrianSoddingBoru4]

That said, let me share with you how I intuit numbers like 2. Based on how I currently see things, there is no number 2 floating out there in the Platonic sense and serving as some form of cause for the concept of 2 in our minds. For me, number 2 strictly exists in our minds, as a way to "visualize" a certain quantity of identical things. The quantity is out there in a "vague" sense, but it is not decipherable as "2" without a mind to see separateness and "identicalness" of the objects of interest. What would be the biggest challenge to this view?

This is all very difficult for me. As so often on this forum, I find myself in the role not of advocating a position but of wanting others to hold back from a position about which they may have too much confidence. I can't say any more than Popper, and his ideas have been worked out in far more detail than I'm aware of, I'm sure. 

One thing to ponder: numbers which no human has ever thought of yet. Imagine, for example, a very large prime number which no one has discovered. It has never appeared in the mind of any person. If numbers depend for their existence on appearing in minds, then this number doesn't exist. But even so, it may be more proper to say that it does exist, but hasn't been discovered yet. If it does exist, but has never been thought, then "where" is it?

But I put the "where" in scare quotes because of our bad habit of thinking of existence in spatial terms. It is begging the question if we say that anything which exists must exist in a location, with extension. This is our habit, especially in modern times, but I'm not sure it's always true. It would probably be better to ask HOW it exists, if it does so without location or extension. 

It may be that while numbers are the inventions of people, they exist not only as their appearance in individual minds. They are somehow commonly held, and exist even if no one is currently thinking them. 

The example Popper uses is about symphonies, which are also World Three objects in his system, like numbers. Imagine Beethoven's 5th symphony. The symphony itself is not identical with its score or its CD. Those are recordings made of it, but are not the symphony itself. It somehow continues to exist whether anyone is hearing it or not. 

I have used the example of Sherlock Holmes, a character everyone knows. Holmes was made up by a person's mind, but is no longer dependent on any individual mind. It may well be that at any given moment, no one in the world is imagining him. Yet he still somehow exists. And though he doesn't exist materially, and can't be the object of scientific studies, it is still possible to make statements about him which are correct or incorrect. And it is possible to distinguish between the "real" Sherlock Holmes ("real" in what way?) and a parody or "re-boot." 

And god knows that no one here wants to talk about theology, but all of this has been discussed for millennia. It is only recent and naive people who assume that the Christian God has a location, physical extension, material existence, etc. As a non-material thingy, it is not the subject of science, but is still said to exist in a way that material things do not.

As a member in good standing of TWO Holmesian societies, I take exception to your characterization of The Great Detective as a fictional character.  This is a base canard promulgated by those lunatics who think Poirot was a better detective. Stupid little Belgie.

But carrying on in the same vein, there are primitive societies in which people cannot conceptualize any number higher than 5 (more than five is 'many'; a lot more than five is 'many many'). There are societies in which large numbers are simply expressions of smaller numbers of groups of numbers.  A fictitious example of the latter is to be found in 'Lord of The Rings' in which Ghân-buri-Ghân express the number 6000 as 'a score of scores counted ten times and five.'


Boru

[Jehanne]

Imagine, for example, a very large prime number which no one has discovered.

Of course, the cardinality of the set of prime numbers is a countably infinite set.  Thousands of mathematical proofs exist that prove such; of course, an acceptance of ZFC and the Axiom of Infinity is necessary to get the balling rolling, so to speak.  Professor Wes Morriston has written extensively in response to WLC, and even quotes Craig who once stated that his ideas will make sense to an individual at least until that person has taken a course in elementary number theory, complex analysis, etc.

[GrandizerII]

Imagine, for example, a very large prime number which no one has discovered.

Of course, the cardinality of the set of prime numbers is a countably infinite set.  Thousands of mathematical proofs exist that prove such; of course, an acceptance of ZFC and the Axiom of Infinity is necessary to get the balling rolling, so to speak.  Professor Wes Morriston has written extensively in response to WLC, and even quotes Craig who once stated that his ideas will make sense to an individual at least until that person has taken a course in elementary number theory, complex analysis, etc.

I keep hearing about this ZFC stuff but never getting around to reading up on it. ELI5 (or perhaps ELI15): What's it about?

---

That said, let me share with you how I intuit numbers like 2. Based on how I currently see things, there is no number 2 floating out there in the Platonic sense and serving as some form of cause for the concept of 2 in our minds. For me, number 2 strictly exists in our minds, as a way to "visualize" a certain quantity of identical things. The quantity is out there in a "vague" sense, but it is not decipherable as "2" without a mind to see separateness and "identicalness" of the objects of interest. What would be the biggest challenge to this view?

This is all very difficult for me. As so often on this forum, I find myself in the role not of advocating a position but of wanting others to hold back from a position about which they may have too much confidence.

It's called revelation!

One thing to ponder: numbers which no human has ever thought of yet. Imagine, for example, a very large prime number which no one has discovered. It has never appeared in the mind of any person. If numbers depend for their existence on appearing in minds, then this number doesn't exist. But even so, it may be more proper to say that it does exist, but hasn't been discovered yet. If it does exist, but has never been thought, then "where" is it?

Yes, it does seem like even numbers that no human being has ever specifically thought of still nevertheless have to exist, but I think that's by nature of being part of the number system which does exist in our minds. That's my first impulse thinking about this question just now.

The example Popper uses is about symphonies, which are also World Three objects in his system, like numbers. Imagine Beethoven's 5th symphony. The symphony itself is not identical with its score or its CD. Those are recordings made of it, but are not the symphony itself. It somehow continues to exist whether anyone is hearing it or not.

I can't say I disagree here. It does seem like abstract objects somehow persist in existence in some unfamiliar way. But I do suspect the [start of] existence of such objects still remains contingent on people's conceptions of them.

But even conceding that, how would you link what you've said here to the original intended topic of this discussion? What does the nature of the existence of numbers suggest regarding the link between mathematics and the physical that challenges the materialistic pov?

[polymath257]

Yes, there is an aspect of math that cuts across cultures. But this is also true of other basic linguistic concepts. So, cat, chat, gato, mao, etc as opposed to two, deux, dos, er. 

The word "cat" and its cognates refers to material objects of a certain type. The question we're working on now is: what does "two" refer to?

What does 'pain' refer to? What does 'abstract' refer to? What does 'love' refer to?

They are *concepts* we have with no external or material referent. The number 2 is like that. It is a placeholder, not a thing in itself.

One difference is that math  is a *formal* language: it has internal rules that are not present in most natural languages. And, for mathematicians, playing with and exploiting those formal rules are the essence of the game.

And, yes, mathematics really is like a very complex game for those doing mathematics. It has rules about what 'plays' are legal, it has goals (theorems), etc. It can even be helpful to *think* of the mathematical concepts visually and in other ways.

You use words that sound unserious when you talk about pure mathematics -- "playing with," "game," "plays," etc. 

Would you say that math is only serious when it is used to describe the material world? That any other time it's just a game? 

I think that many mathematicians would disagree with you. I'm also concerned that if we define seriousness as utility, we're repeating a common anti-intellectual assumption.

In case you haven't noticed, people can take games very seriously.There are serious chess players, serious athletes, etc. That doesn't mean they aren't ultimately playing a game.

Mathematics is a very challenging intellectual game. It has puzzles in it, challenges, exercises, etc. But in and of itself it says NOTHING about reality, except that the game can be played.

Now, it does turn out to be a *useful* game, but like you said, that is a different matter. Whether that utiliy is serious or not depends on the user.

In exactly what sense do numbers have an 'independent existence'? 

Since you've already ruled out all of Plato (without bothering to explain why), we can talk about numbers having independent existence in exactly the way that Popper describes. I suppose I could type that all out in my own words, but the original paper is not long. And if you're going to go around declaring that Popper is wrong it might make sense for you to read what he says. 

https://tannerlectures.utah.edu/_documen...pper80.pdf

What I find most interesting is that Popper *doesn't* claim that world 3 objects have an independent existence. They are created by humans for human reasons and to do things to and for humans.

And, like the world 2 objects, I think they ultimately supervene on the physical. They are *concepts* in human minds, common actions or beliefs across humans. They only exist because we exist thinking about them.

From what I can see, the 'number 2' is a shorthand for all the cases where counting two objects is a useful thing to do. And the mathematical object 2 allows for such modeling.

So a number is the memo we use after looking at two objects and counting them? If that's all numbers are, then you're begging the question and assuming that they only exist when used in reference to physical objects. 

No, I am saying they only exist as concepts in our minds. They have no independent reality. We can, for example, use them to count thoughts or imaginary pink unicorns.

But then, you use the term "mathematical object 2." Is this the same as the number 2? Do people doing pure mathematics use "mathematical objects" but not numbers?

I would guess that very few ordinary people think of the number 2 as the set {{},{{}}}. They are more inclined to think of 2 as being the successor of 1 in some inductive set, but I doubt they think of it in those terms.

Both of the mathematical formulations are based in the specific axioms of set theory used for the development of modern mathematics.

And, in most ways, the second sense is the closest to an actual definition of 2. You start with an inductive set: just a set with a one-to-one function and a special point (depending on the formulation, we call that special point 0 or 1). if we call the special point 0, we define 1 to be the image of 0 under that function. And, in either case, we define 2 to be the image of 1.

But what this means is that there are *many* versions of the number 2: one for each inductive set. They are considered equivalent because equivalences between such inductive sets.

The average mathematician would find this rather obvious after a bit of thought. But I very highly doubt that the average person has ever thought in these terms.

In other words, there *is no number 2* that is different from every other mathematical object. The property of being 2 depends on the specific system it is in (the inductive set). And it is a matter of definition (does our inductive set start with 0 or 1?)

Now, the first definition of 2, as {{},{{}}}, is useful in math because it gives a very simple example of a set with the required properties for a very easily defined inductive set (the ordinals). But, again, I very much doubt that the average person thinks like that.

On the other hand, Frege's definition of 2 as being the collection of all sets of pairs turns out to be inconsistent. That collection cannot be a set because of Russell's paradox (ultimately).

---

I really want to see some one counting in 'Cat'!

  

No one counts in "cat." 

The question is: does a word like "two" refer to an object in the same way that a word like "cat" does.

No, it doesn't. But neither do words like 'thought' or 'good' or 'beauty'.

[The Grand Nudger]

Thoughts refer to an object, we're just not used to thinking of them that way because the mechanism which produces them doesn't communicate the specifics of their creation and maintenance to the apparatus which directs attention.

Us.

Good and beauty, likewise, objects.

[polymath257]

Thoughts refer to an object, we're just not used to thinking of them that way because the mechanism which produces them doesn't communicate the specifics of their creation and maintenance to the apparatus which directs attention.  

Us.

Good and beauty, likewise,  objects.

I actually bet that 'thought' refers to a *process*, not an object in the brain.

[UtilitarianDeist]

I agree that thoughts are objects. I also think that subjects are objects. A type of object.

I disagree that processes are not objects. To me a process is just a type of thing and an object is the same thing/same object as a thing/object. Everything is a thing because things are objects and every object is an object. i use object and thing interchangeably. You may have noticed.

I mean, you could say that a process is not an object because it's a set of objects. But the point is that it's still about objects. Thoughts are objects indeed. Thoughts are also subjects ... but there's no contradiction there, actually.

[The Grand Nudger]

The state of that process is the object of reference. The product of the state is useful to us, but knowledge of the object or process itself may not be.

We don't need to know the physical coordinates of the thought "run!" for it to be useful, but it does have coordinates. A crude metaphor would be the actual list of circuits or even individual gates used in any given product of mechanical computation. A full description could even include the energy used at those coords.

[UtilitarianDeist]

The state of that process is the object of reference.  The product of the state is useful to us, but knowledge of the object or process itself may not be.

We don't need to know the physical coordinates of the thought "run!" for it to be useful, but it does have coordinates.  A crude metaphor would be the actual list of circuits or even individual gates used in any given product of mechanical computation.  A full description could even include the energy used at those coords.

Good point. And just because the sun is made of smaller objects doesn't mean that the sun isn't an object. The sun can be both an object and a set of smaller objects.

Just as with a process. A process can be both an object and a set of smaller objects that working together we call a 'process'.

So, when I said "you could say that a process is not an object because it's a set of objects." I was wrong. I retract that statement.