Do Mathematical Entities Really Exist?

[Kaylee]

Do mathematical entities really exist? Wowbagger's thread "Why is pi an irrational number" diverged at times into this topic. I found this former math professor's (@ University of Hawaii) essay on the subject
( http://www.math.hawaii.edu/~lee/exist.html )
pretty cool. It includes ramblings on:

• Do numbers (even the plain 'ol "natural numbers" ) really exist, or are they just a human construct?
• Does infinity really exist?
• Numbers sets
• Some nice short explanations about calculus
• Some history
• And much more -- however, unfortunately, the number 42 isn't discussed.

 

[nimzov]

Do mathematical entities really exist? Wowbagger's thread "Why is pi an irrational number" diverged at times into this topic. I found this former math professor's (@ University of Hawaii) essay on the subject
( http://www.math.hawaii.edu/~lee/exist.html )
pretty cool. It includes ramblings on:

• Do numbers (even the plain 'ol "natural numbers" ) really exist, or are they just a human construct?
• Does infinity really exist?
• Numbers sets
• Some nice short explanations about calculus
• Some history
• And much more -- however, unfortunately, the number 42 isn't discussed.

If they (numbers and their mathematical relations) do not exist somewhere in the universe (I mean outside our brain), it must be an extraordinary coincidence that they can "fit" so well to our universe ?

nimzo

 

[drkitten]

Do mathematical entities really exist? Wowbagger's thread "Why is pi an irrational number" diverged at times into this topic. I found this former math professor's (@ University of Hawaii) essay on the subject
( http://www.math.hawaii.edu/~lee/exist.html )
pretty cool.

Indeed pretty cool -- but it (fortunately or unfortunately) also answers your question within the first two paragraphs. "It depends on what you mean by `exist.'" As he pointed out, Frodo Baggins is fictional (does not exist), but in the context of a literary discussion of the Lord of the RIngs, frodo exists, but MacGruff the Crime Dog doesn't.

From here on out one starts the inevitable dribbling into philosophical word games and semantic quibbles.....

 

[drkitten]

If they (numbers and their mathematical relations) do not exist somewhere in the universe (I mean outside our brain), it must be an extraordinary coincidence that they can "fit" so well to our universe ?

I doubt it. Any more than it's an extraordinary coincidence that all of the drawings I've seen of Minas Tirith "fit" so well to the laws of perspective that I'm used to in our universe.

 

[nimzov]

Indeed pretty cool -- but it (fortunately or unfortunately) also answers your question within the first two paragraphs. "It depends on what you mean by `exist.'" As he pointed out, Frodo Baggins is fictional (does not exist), but in the context of a literary discussion of the Lord of the RIngs, frodo exists, but MacGruff the Crime Dog doesn't.

From here on out one starts the inevitable dribbling into philosophical word games and semantic quibbles.....

If you don't like "exist" then we could ask : is this true outside the solar system ? Or was it true 10 millions years ago ? Or will it still be true tomorrow?

nimzo

(I must say I haven't yet read the article)

 

[drkitten]

If you don't like "exist" then we could ask : is this true outside the solar system ? Or was it true 10 millions years ago ? Or will it still be true tomorrow?

I guess we could, but I don't see the relevance. If I don't like "truth," either, does that mean I get to ask : Does pasta puttanesca taste better or worse if you leave out the anchovies?

(As it happens, I'm not sure that your sentence is actually "true." What happens when you evaluate it in the field of integers modulo 2? I believe the infinite sum diverges -- while of course the closed form expression doesn't even have an interpretation....)

 

[nimzov]

I guess we could, but I don't see the relevance. If I don't like "truth," either, does that mean I get to ask : Does pasta puttanesca taste better or worse if you leave out the anchovies?

This Euler relation is true or it is not true. I believe it is equaly correct for us as for hypothetical aliens. I guess mathematical abstractions have a better predictive record for how the world works than pasta.

nimzo

 

[drkitten]

This Euler relation is true or it is not true.

.... or both, or neither.

As I pointed out above, it's true in the field of real numbers, false in the field of Z/2Z.

But that says nothing about whether or not the Euler relation exists. The statement that Bilbo Baggins is Frodo's uncle is either true or false, but that doesn't mean that either Bilbo or Frodo -- or their hypothetical relationship -- exists.

 

[Ashles]

• Do numbers (even the plain 'ol "natural numbers" ) really exist, or are they just a human construct?
• Does infinity really exist?
• Numbers sets
• Some nice short explanations about calculus
• Some history
• And much more -- however, unfortunately, the number 42 isn't discussed.

Oddly enough I was wondering about this the other day when looking at a current Burger King advert.

It claims their Whopper has "As much beef as is mathematically possible".

This got me thinking. Does that make any sense? As much beef as physically possible would make sense, as it is conforming to rules that we have discovered are inbuilt into the universe.
But isn't maths just a tool for our analysis of the universe rather than an actual set of rules that we have discovered?

What would be a mathematically impossible amount of beef?

 

[Kaylee]

Indeed pretty cool -- but it (fortunately or unfortunately) also answers your question within the first two paragraphs. "It depends on what you mean by `exist.'" As he pointed out, Frodo Baggins is fictional (does not exist), but in the context of a literary discussion of the Lord of the RIngs, frodo exists, but MacGruff the Crime Dog doesn't.

I found his intro very glib and the rest of what he had to say more satisfying even if he does ramble. He does come back to that particular issue again and again.

He dismisses his own intro with:

…they [numbers, my edit] are not arbitrary products of our imaginations in the way that fictional characters are.

but still alludes to the problem of [cultural] definitions with:

One of the sources of confusion, as I see it, is that in mathematics we think of numbers as nouns. But if we trace the concept down to its roots, we see that, in origin, numbers are adjectives. (Footnote: This statement is colored by the fact that my native language is English. I know that in Japanese, for instance, colors are fundamentally nouns. So perhaps in that language numbers are also nouns.) And as adjectives, one does not have the same dispute as to whether numbers exist or not. There is no question that in the world there do exist collections of five objects. And there exist such things as a half liter of beer or half a quart of milk.

Thus the question as to whether natural numbers and simple fractions exist is like the question as to whether red exists. There is no doubt but whether there exist red objects. Red as a color, on the other hand, is not a tangible object. It is a linguistic construct. Nonetheless, red does exist. For most people, anyway, a statement that red is fictional would be considered simply crazy.

and discusses the issue of practical examples:

In any case, this question of the existence of numbers is a philosophical one, i.e. one of no practical importance. Certainly almost no contemporary mathematician would think that this question has any relevance to mathematics. And yet it does come up in the teaching of mathematics when students ask, for instance, whether there really exists a square root of minus one. And when the teacher answers, as I usually did, "No. We made it up, just like we made up all the other numbers," students find this a little disconcerting.

When students ask whether negative numbers actually exist, teachers frequently cite the examples of going into debt or negative temperatures on a thermometer. In doing this, one is basically showing the student that negative numbers function as adjectives describing the tangible world. But when it comes to the square root of minus one, it is almost impossible to come up with comparable examples.

...
<snip>
...

"Find [f(x+h) - f(x)] /h if f(x) = x² - 3x + 12."

I often had students come to me in bewilderment, saying, "I have no idea what this problem means." The increase in level of abstraction is more rapid than many students can deal with.

<snip>

The same student who is so bewildered by "Find [f(x+h) - f(x)] /h" is likely to find that everything becomes clear when he learns that the derivative of distance is velocity and the derivative of velocity is acceleration.

He discusses the idiom issue also, that is, common words have different meanings in different fields and in mathematics definitions can even vary depending upon which theory you are speaking about. He starts discussing that issue under the heading:
Formal Definition of the Numbers Concept and goes on for a few more sections.

His quotes Kronecker (even though he disagrees with his viewpoint):

The natural numbers were created by God; all the others are the invention of humans.

and introduces the "Problem of Universals":

the question as to whether abstract concepts have some sort of real existence in the world, or whether they exist only in our minds. Like most philosophical problems, it seems to be more a question about language than a question about the world, although there are certainly philophers who would disagree with me in this respect.

From here on out one starts the inevitable dribbling into philosophical word games and semantic quibbles.....

Dribbles perhaps. But sometimes these meanderings help clarify our understanding of the world. I might not suggest that most college students major in philosophy, but reading about it can be illuminating. I would find it really amusing if a university professor like yourself thinks otherwise, but then again I do get easily amused sometimes.

Anyway, back to my OP's link, I think his essay is well worth reading past the first two paragraphs. I think its even worth reading all the way through.

 

[nimzov]

What would be a mathematically impossible amount of beef?

101% beef.

.... or both, or neither.

As I pointed out above, it's true in the field of real numbers, false in the field of Z/2Z.

You can use if you like another mathematical expression that is true for the field Z/2Z. So my question remains : are true mathematical relations true everywhere in the universe ?

But that says nothing about whether or not the Euler relation exists. The statement that Bilbo Baggins is Frodo's uncle is either true or false, but that doesn't mean that either Bilbo or Frodo -- or their hypothetical relationship -- exists.

Statements that have no predictive calue in the universe are not comparable, in my opinion to mathematical objects.

Can mathematical relations that are true not exist ? I don't know.

nimzo

 

[Kaylee]

Oddly enough I was wondering about this the other day when looking at a current Burger King advert.

It claims their Whopper has "As much beef as is mathematically possible".

This got me thinking. Does that make any sense? As much beef as physically possible would make sense, as it is conforming to rules that we have discovered are inbuilt into the universe.
But isn't maths just a tool for our analysis of the universe rather than an actual set of rules that we have discovered?

What would be a mathematically impossible amount of beef?

Infinity.

That's a strange advert. -- but since you remembered it I guess the ad team did its job.

FWIW, I agree with the folks who say mathematics is a language that we use to discover the rules of the universe but I got my BS (pun intended) in accounting, so I'm not sure my opinon in this area carries any weight with anyone other than myself...

 

[drkitten]

You can use if you like another mathematical expression that is true for the field Z/2Z. So my question remains : are true mathematical relations true everywhere in the universe ?

Your question makes no sense. If the mathematical relationship you describe is neither true nor false, why would swapping seats at the dinner table make any difference?

 

[politas]

What would be a mathematically impossible amount of beef?

5.9742 × 10²⁴ kilograms would certainly be impossible until we get extraterrestrial beef production happening.

 

[nimzov]

Your question makes no sense. If the mathematical relationship you describe is neither true nor false, why would swapping seats at the dinner table make any difference?

I don't see why me saying that the Euler relation is true for the natural numbers makes no sense ?

nimzo

 

[drkitten]

Anyway, back to my OP's link, I think his essay is well worth reading past the first two paragraphs. I think its even worth reading all the way through.

Oh, I agree. His -- very glib -- first two paragraphs are simply trying to illustrate that the only simple answer is "if you can ask that question, you don't have a good understanding of what you mean by `exists.'"

The rest of his essay is him trying to explain what he means by "exists." And it's a very good explanation, for what it's worth -- but, as is so often the problem in philosophy, it's not necessarily an explanation or a meaning that will be shared among everyone who reads it. Many will even read it and reject his explanation and meaning out of hand, without articulating their own, simply because they disagree with his conclusion. ("I may not know what a right answer looks like, but I know that the one you've presented is wrong -- and therefore your premises are equally wrong." I've seen that response a lot in philosophy classes. Usually just before the teacher unleashes the mighty Hammer of Sarcasm that you may have seen in The Wall.)

 

[drkitten]

I don't see why me saying that the Euler relation is true for the natural numbers makes no sense?

For a start, because pi is not defined in "the natural numbers."

For a followup, because "the natural numbers" do not of themselves impose an interpretation of symbols such as +.

 

[andyandy]

interesting question.....

my own opinion would be that mathematics is the underlying reality of the universe.
If the universe is completely deterministic then i think that the all encompassing theory of everything would be that which really exists - the universe as we percieve it exists as a representation of that underlying reality - a bit like how the underlying reality of this website is its source code and that which we perceive as existing is just a manifestation of that code.

lol
although i'll admit i'm way out of my depth on this one (not that it usually stops me giving an opinion )

 

[Kaylee]

Oh, I agree. His -- very glib -- first two paragraphs are simply trying to illustrate that the only simple answer is "if you can ask that question, you don't have a good understanding of what you mean by `exists.'"

The rest of his essay is him trying to explain what he means by "exists." And it's a very good explanation, for what it's worth -- but, as is so often the problem in philosophy, it's not necessarily an explanation or a meaning that will be shared among everyone who reads it. Many will even read it and reject his explanation and meaning out of hand, without articulating their own, simply because they disagree with his conclusion. ("I may not know what a right answer looks like, but I know that the one you've presented is wrong -- and therefore your premises are equally wrong." I've seen that response a lot in philosophy classes. Usually just before the teacher unleashes the mighty Hammer of Sarcasm that you may have seen in The Wall.)

For some reason I missed that movie, but interesting metaphor.

 

[Darth Rotor]

Indeed pretty cool -- but it (fortunately or unfortunately) also answers your question within the first two paragraphs. "It depends on what you mean by `exist.'" As he pointed out, Frodo Baggins is fictional (does not exist), but in the context of a literary discussion of the Lord of the RIngs, frodo exists, but MacGruff the Crime Dog doesn't.

From here on out one starts the inevitable dribbling into philosophical word games and semantic quibbles.....

Which is why the internet was invented for porn, not forums.

I actually liked your post, just could not resist. The spirit is willing, but the flesh is weak.

DR