Mathematics

[lpetrich]

Philosophy of mathematics^WP mentions several schools of thought about this issue.

These may broadly be divided into

• Realism: mathematical entities exist independent of us -- Platonism, empiricism
• Anti-realism: mathematical entities are figments of our imaginations -- formalism, fictionalism

More detail:

• Platonism - there is a special realm of mathematical entities
• Empiricism - mathematics as an experimental science
• Mathematical monism - Max Tegmark's hypothesis that the Universe is made of mathematics
• Logicism - mathematics is derived from logic
• Formalism - mathematics is a consequence of various string-manipulation rules
• Conventionalism - mathematics is a social convention
• Psychologism - mathematics is a result of our psychology
• Intuitionism - only experienced mathematical entities are valid
  • Constructivism - only constructed mathematical entities are valid
  • Finitism - constructivism, with a finite number of steps from the natural numbers
  • Ultrafinitism - finitism, limited by available resources
• Structuralism - mathematical entities are structures
• Embodied mind theories - mathematics a result of our
  • New empiricism
  • Aristotelian realism
• Fictionalism - mathematics as a convenient fiction
• Social constructivism or social realism - mathematics as a social construct

 

[bob_kark]

So…the question (so far unanswerable) is: What is the relationship between mathematics and the universe in which it occurs? There has to be one. At the very least there must be some way of explaining how a brain generates it (given the current state of neuroscience…that understanding is likely decades away at best). But…there also HAS to be some explanation for why we find mathematics….everywhere.

I think the question you should be asking is "why is the universe sometimes predictable?" I think the answer you're looking for lies outside the world of mathematics and is more philosophical than scientific in nature, given our current level of understanding. Though there are scientific debates of this very topic, they're inconclusive, to my understanding. The most common being, if the universe were unpredictable, there would not be a suitable environment for us to exist, much less ask the question. Though I'm sure there may be others on this forum that may be capable of proving my statement regarding the nature of the answer to this question to be wrong.

 

[Dessi]

Obviously…1+1=2 is a property of the universe…because it is created by a brain that itself is a function of the universe. So far, no one has a clue what the relationship between the brain and 1+1=2 actually is.

I take 1 apple and put it on an empty table, and there are 1 apples on the table.
I take 1 apple and put it on the table with the other apple, and there are 2 apples on the table.
QED.
Also, apples.

 

[pgwenthold]

Obviously…1+1=2 is a property of the universe….

1 cup of popcorn + 1 cup of water does not give you two cups of soggy popcorn.

1 + 1 = 2 is not a universal property, and only works for properly defined circumstances.

 

[annnnoid]

Logic and mathematics are manifested as the fundamental nature of the universe. The evidence for this is overwhelming. Logic and mathematics are not invented by humans; they are discovered. The vocabulary and language referred to above are the tools invented for this discovery. Let's not confuse the two.

So far no one anywhere has explicitly identified the fundamental ontological reality of the universe. Not even close (it may ‘look’ mathematical…but jumping from that to ‘it is math’ is …to borrow a mathematical term…an all-but infinite distance). Therefore…in order for you to be able to prove that logic / math are discovered you would have to be able to explicitly establish the mechanism by which the human brain creates consciousness (…and thus creates logic / mathematics). At present nobody has the slightest idea how consciousness is generated by the physical activity of the brain. Therefore your claim is fundamentally flawed. Sorry.

Philosophy of mathematics^WP mentions several schools of thought about this issue.

These may broadly be divided into

(smip)….

Very useful stuff. Interesting how one vocabulary adjudicates another vocabulary.

I take 1 apple and put it on an empty table, and there are 1 apples on the table.
I take 1 apple and put it on the table with the other apple, and there are 2 apples on the table.
QED.
Also, apples.

1 cup of popcorn + 1 cup of water does not give you two cups of soggy popcorn.

1 + 1 = 2 is not a universal property, and only works for properly defined circumstances.

I didn’t say it was an axiomatic property (though there is some reason to believe there are such things). It is merely somehow created by this universe (obviously…because it exists so there must be some kind of relationship between whatever-the-universe-is and 1+1=2).

I think the question you should be asking is "why is the universe sometimes predictable?" I think the answer you're looking for lies outside the world of mathematics and is more philosophical than scientific in nature, given our current level of understanding. Though there are scientific debates of this very topic, they're inconclusive, to my understanding. The most common being, if the universe were unpredictable, there would not be a suitable environment for us to exist, much less ask the question. Though I'm sure there may be others on this forum that may be capable of proving my statement regarding the nature of the answer to this question to be wrong.

That is simply becoming too abstract.

 

[Perpetual Student]

So far no one anywhere has explicitly identified the fundamental ontological reality of the universe. Not even close (it may ‘look’ mathematical…but jumping from that to ‘it is math’ is …to borrow a mathematical term…an all-but infinite distance). Therefore…in order for you to be able to prove that logic / math are discovered you would have to be able to explicitly establish the mechanism by which the human brain creates consciousness (…and thus creates logic / mathematics). At present nobody has the slightest idea how consciousness is generated by the physical activity of the brain. Therefore your claim is fundamentally flawed. Sorry.

<...>

There is nothing in science that is "proven." The universe is demonstrated to be logical and mathematical in the same sense that every electron is demonstrated to have the same electric charge of 1.60217657 × 10-19 coulombs. Neither statement can be proven, but we have overwhelming evidence to support both statements.
That the universe would behave the same logical and mathematical way without human intelligence or consciousness is evidenced by the fact that it existed in exactly that manner for 13+ billion years. We have been here a very short time to make these discoveries about logic and mathematics. No mechanism regarding consciousness is necessary to make that claim.

 

[psionl0]

1 cup of popcorn + 1 cup of water does not give you two cups of soggy popcorn.

No, but you now have two objects.

If you object to that description then call it two non-empty cups.

 

[Almo]

I had a bad feeling about this thread given the title.

 

[Planigale]

To put a 'quantum' spin on this. Mathematics is the study of …. This requires an observer to study. In the absence of an observer there is no mathematics. Things happen they are only mathematical when observed, and a relationship drawn. To put a zen spin 1 apple plus one apple may make two apples put in the absence of an observer are there any apples? Apples are just an energy pattern that humans define as an apple.

 

[Dessi]

Is this official math crackpot thread?

 

[psionl0]

This requires an observer to study.

It doesn't really. It is just an application of logic.

To un-zen it a little,
IF an apple is placed into an empty region of space AND another apple is placed into the same region
THEN there are two apples in that region of space.

Now it doesn't matter if these actions were observed nor even if it never happened at all. It is a conditional statement which is logically true.

 

[psionl0]

Is this official math crackpot thread?

One crackpot does not make a pile of broken pottery.

 

[Soapy Sam]

Mathematics is a discipline which exists entirely independent of
conception for while the symbols may be human it itself is not
But what do the honourable members here think of this though

Whether this is nonsensical or not depends on your definition of "conception".
If a concept can only exist in a human brain , as I would take to be self evident, then mathematics is an entirely human neural process, like any other grammar. That is, of course, a circular argument, but any alternative supposes some vague external reality for mathematics. I doubt you would propose the same for language.
If you mean mathematics exists in some sense independently of human minds, then perhaps you will clarify the nature of that mechanism?

 

[marplots]

It doesn't really. It is just an application of logic.

To un-zen it a little,
IF an apple is placed into an empty region of space AND another apple is placed into the same region
THEN there are two apples in that region of space.

Now it doesn't matter if these actions were observed nor even if it never happened at all. It is a conditional statement which is logically true.

Does it matter if the experiment is even possible? Because when I try it, I always get something other than two apples, with answers like: 2.15, 1.78 and so on.

 

[psionl0]

Does it matter if the experiment is even possible? Because when I try it, I always get something other than two apples, with answers like: 2.15, 1.78 and so on.

That's funny, I always get 2.

But no, if the experiment is impossible (ie the premise is false) then the conditional statement is still true.

 

[marplots]

That's funny, I always get 2.

But no, if the experiment is impossible (ie the premise is false) then the conditional statement is still true.

Just not true in this universe.

 

[psionl0]

Just not true in this universe.

The conditional statement is true in any universe (unless you can provide a counter example).

This is the table of truth values for the experiment:
[table=head]Experiment can be conducted|Answer=2|Conditional Statement is true
false|false|true
false|true|true
true|false|false
true|true|true[/table]

 

[marplots]

The conditional statement is true in any universe (unless you can provide a counter example). (First and second row.)

This is the table of truth values for the experiment:
[table=head]Experiment can be conducted|Answer=2|Conditional Statement is true
false|false|true
false|true|true
true|false|false
true|true|true[/table]

I'm sure I just misunderstand your meaning, but in your truth table, if the experiment is impossible, how do you get any answer at all in the second column?

Say we changed your syllogism to conclude something like:
Leprechauns can ride unicorns.

Would your statement: "But no, if the experiment is impossible (ie the premise is false) then the conditional statement is still true" still apply?

In other words, don't conditionals depend on the conditions not being nonsensical? Otherwise, the whole exercise seems to be a kind of begging the question. But, as I said, it is more likely I just don't understand something about how you are expressing the assertion.

 

[pgwenthold]

No, but you now have two objects.

If you object to that description then call it two non-empty cups.

No and no.

I have one "object" - soggy popcorn

And there are lots of ways to do it where I don't have two empty cups.

 

[psionl0]

.............................
If you object to that description then call it two non-empty cups.

.........................
And there are lots of ways to do it where I don't have two empty cups.

How did two non-empty cups suddenly become two empty cups?