Do Mathematical Entities Really Exist?

[l0rca]

I don't remember if it was you Dave, but I think it was, who last time I was here was entertaining a number of General Relativity questions. That would make you the math guy.

 

[Kaylee]

I'm in the unenviable position of not really being able to contribute much to a thread that I started.

Ah well. I'll put in my few cents anyway and as for the rest I'll bring out the popcorn and watch.

How do you think the limits of our brains play into that? The question of whether it's underlying reality we're discovering, or the fundamental limiting structures of our brains.

How about -- We discover the underlying reality despite the constraints of our brain, senses, and physical limitations of our bodies. (If my cat had a pair of opposable thumbs and a handful of flexible fingers -- with no other physiological changes -- she would be quit a formidable individual. Well OK, I'm exaggerating for the sake of a more interesting post -- but I'm not exaggerating by much. )

... But, I think it treads into philosophical realms, where multiple people could come up with different reasonable definitions of "exists", and although some of those definitions might conflict with each other, they would still be correct, from a particular point of view. I think Daniel C. Dennett wrote a book about such things, he refers to as the "intentional stance". You could claim that Santa Claus exists, if by "Santa Claus" you were referring to some particular guy walking on the street, even if said person does not look anything like "Santa Claus" as most others would define him.

Does that make sense, or is that another example of rambling?

In my humble view, it seems mathematics evolved as a very concrete method of defining things, and solving problems. Mathematics may not be very real, as far as the rest of the natural Universe is concerned, but for us, it is an undeniably useful tool for examining the world.

In my pi thread, someone brought up the idea that numbers are like adjectives, used to describe things, like "ugly" is a verb, and thus numbers do not really exist as physical entities, but rather descriptions of entities. I think the biggest difference between "ugly" and, say, the number 2, is that "ugliness" is subjective: Two people can disagree on what is ugly, and both be right. Yet, the number 2 is objective: If two different people disagree on how many things is 2, then one of them is probably crazy.

So, to summarize, mathematics is about the most realistic method we have for modeling our Universe, but the numbers, themselves, do not exist as entities: They are merely convenient states of thinking we evolved to help us live.

The author sounds interesting and I added his name to my reading list -- thanks.

Re difference between "ugly" and "2" -- great insight. But because it's so unlikely for people to disagree on how many things are two -- doesn't that help support the position that numbers are a concrete reality and not an abstraction?

I believe that infinity is abstract and that our ability to enumerate into the quintillions is enabled by the development of the value placement system, which required abstract thought.

But even monkeys recognize that 4 bananas are more than 2. I think they are recognizing a concrete reality. I think our difference of opinion is due to where we believe concrete reality stops and abstraction begins.

Perhaps one sign of where this point begins is when subjectivity starts to come into play. I'm not a mathematician -- I always thought an equation either works or it doesn't. However after reading Mercurytorrents's post, I'm not sure that we (the posters in this thread) could come to an agreement as to which specific idea, axiom or theorum should be targetted to show where subjectivity in mathematic starts.

Math originally was, and in real number systems, still is a game of being specific. This math in theory takes quantities in supposed equal sums and computes them. Symbols see. These symbols perfectly represent reality, again in theory, because they carry no meaning beyond quantity, and the theory assumes quantities do not randomly change, so long as they are initially accurate.

So this sort of math is a subjectional [ed: I added the bolding here and in the next one also ] reference point for real quantities. It's subjectional because this human invention of math has its own individual connotations of emotion and history with its users.

So far as math as an entity, no, it does not exist at all. The universe does not operate according to mathematical principles. It can be described in math as it can in the best of words, though perhaps better, since math relays concrete quantities quite bare to mind, though our minds do dress them so. Still, the workings of the universe are unknown.

Now if we were to redefine math not as a computational ideal, but as an ambiguous, simple set of underpinnings for which things seem to operate, inversely identifiable by complex algorithms and the sort, we could say yes, math as an entity exists. But why cause such pejoration when we already have such a good grasp of our own subjectivity?

 

[Zygar]

I suppose we don't have any hardcore mathematicians here?

Not that I enjoy math much. But I'll describe here, as simple logic compounded, and eventually in its use morphed, morphed enough to draw the line.

Math originally was, and in real number systems, still is a game of being specific. This math in theory takes quantities in supposed equal sums and computes them. Symbols see. These symbols perfectly represent reality, again in theory, because they carry no meaning beyond quantity, and the theory assumes quantities do not randomly change, so long as they are initially accurate.

So this sort of math is a subjectional reference point for real quantities. It's subjectional because this human invention of math has its own individual connotations of emotion and history with its users.

So far as math as an entity, no, it does not exist at all. The universe does not operate according to mathematical principles. It can be described in math as it can in the best of words, though perhaps better, since math relays concrete quantities quite bare to mind, though our minds do dress them so. Still, the workings of the universe are unknown.

I think that the only way I could accept the idea that mathematical entities do not exist is if we can also say that the laws of physics do not exist. I would agree that math in and of itself is definitely too abstract an idea to exist. But the entities used by mathematics to understand and describe our universe are very real.

Mathematics are definitely able to describe, objectively, a number of very abstract concepts. Just because humans created a slew of symbols and ways of organizing said symbols does not mean that the underlying mathematics of those symbols does not describe the universe. Infact they do. And as long as the laws of physics exists, the universe is adhering to these laws. Which means that the universe does operate according to mathematical principles.

Now if we were to redefine math not as a computational ideal, but as an ambiguous, simple set of underpinnings for which things seem to operate, inversely identifiable by complex algorithms and the sort, we could say yes, math as an entity exists. But why cause such pejoration when we already have such a good grasp of our own subjectivity?

I simple do not understand this statement. Mathematics are not defined as a computational ideal. Seeing it as simple a computational ideal is leaving out the primary foundation of mathematics, which is proofs. These proofs are themselves provable in any language to any being capable of grasping its conceptual framework. They exist with or without our intervention, but understanding them is the foundation of mathemetics.

 

[Blert]

In my humble view, it seems mathematics evolved as a very concrete method of defining things, and solving problems. Mathematics may not be very real, as far as the rest of the natural Universe is concerned, but for us, it is an undeniably useful tool for examining the world.

In my pi thread, someone brought up the idea that numbers are like adjectives, used to describe things, like "ugly" is a verb, and thus numbers do not really exist as physical entities, but rather descriptions of entities. I think the biggest difference between "ugly" and, say, the number 2, is that "ugliness" is subjective: Two people can disagree on what is ugly, and both be right. Yet, the number 2 is objective: If two different people disagree on how many things is 2, then one of them is probably crazy.

So, to summarize, mathematics is about the most realistic method we have for modeling our Universe, but the numbers, themselves, do not exist as entities: They are merely convenient states of thinking we evolved to help us live.

Hear! Hear! This is pretty well put; however, I disagree that the number 2 is objective---the equivalence class of all sets with the same cardinality as the power set of the power set of the empty set is objective: calling this critter "2" is merely a local prejudice. (*wink, wink*)

 

[Blert]

I think that the only way I could accept the idea that mathematical entities do not exist is if we can also say that the laws of physics do not exist. I would agree that math in and of itself is definitely too abstract an idea to exist. But the entities used by mathematics to understand and describe our universe are very real.

Nah, actually you've got it all wrong (and I am a hardcore mathematician, or at least a close approximation thereof). I think you may be confusing the actual laws of physics with our mathematical descriptions of what we think are the laws of physics (check back in about 300 years, and I think you'll find we've had it all wrong---that is to say, by then we'll have come up with mathematical descriptions that better match what we think are the laws of physics).

Mathematics is often inspired by real-world objects, but mathematicians *never* works with real-world objects, *always* with abstractions that try to involve the relevant properties of the objects without any of the fiddling details that are unnecessary. This is actually what most people find difficult about mathematics: mathematicians try to abstract the problem to the point that it doesn't actually matter what you apply the results to. This is one of the things that makes it rather difficult to learn, but very powerful to apply.

Take, for instance, the problem of finding the height of a tree from the length of its shadow and the angle of the sun. A pure mathematician only sees the right triangle with the given base and angle---never mind that that triangle does not actually exist (nor, really, can it, other than as a picture in our minds). The mathematician studies the triangle; it matters little to him whether you're applying the trig to find the height of a tree, or the height of the top of a ladder leaning against a wall, or the y-component of a vector given the x-component and the direction, or any number of other 'real-world' applications you can find.

In fact, he's more likely to be fascinated by this idealized triangle and will almost certainly wonder not what he can do with this triangle, but why it works. If he has a philosophical bent, he'll trace the ideas back to the basic assumptions that are being made in coming up with the solution to the problem and begin tinkering with those assumptions to see how changing them changes the final outcome.

Mathematics are definitely able to describe, objectively, a number of very abstract concepts. Just because humans created a slew of symbols and ways of organizing said symbols does not mean that the underlying mathematics of those symbols does not describe the universe. Infact they do. And as long as the laws of physics exists, the universe is adhering to these laws. Which means that the universe does operate according to mathematical principles.

It would probably be more accurate to say that the mathematical principles work the way they do because we are trying to describe how the universe operates, not the other way around. If the mathematics does not predict the universe, we change some definitions or assumptions until it does. If, in addition, the model predicts something that we hadn't known before, scientists rush out and test it---if the model does actually predict something and it turns out to be true, it lends the the mystique that the universe is somehow mathematical in nature; if what is predicted does not turn out to be true, we begin tinkering with the model until it agrees with what we observe.

In a sense, this mystique is sort of like the mystique some folks have regarding the power of prayer---if they get what they pray for, they believe more strongly in the power of prayer; if they don't, they figure they didn't deserve it in the first place.

I simple do not understand this statement. Mathematics are not defined as a computational ideal. Seeing it as simple a computational ideal is leaving out the primary foundation of mathematics, which is proofs. These proofs are themselves provable in any language to any being capable of grasping its conceptual framework. They exist with or without our intervention, but understanding them is the foundation of mathemetics.

Proofs, while the primary tool of mathematicians, are fundamentally dependent on the axioms you start with---that is to say, what assumptions you make. Philosophically, mathematicians like to work with as few assumptions as possible, and prove as much as we can from those assumptions. But change one little thing in any of the basic axioms, and you'll find that any proof that depended on that axiom is no longer valid in the new axiom scheme. In fact, you usually get a new set of proofs and a new set of results (with some commonalities to the old system, if you didn't change too many of the basic assumptions). The mathematics is correct in each of the schemes, but some may be more applicable to the 'real world' than others.

Look up some information on Non-Euclidean geometries if you're curious about this.

 

[l0rca]

Thanks for the well done post, Blert.

I'd might have replied myself, but I was terribly drunk last night.

 

[Roboramma]

If you mean by mathematical entities existing that, for instance, if you have 2 of something, and you get 2 more of those things, you'll have 4 of them, then obviously they exist. And I think this is true of all of the more complex cases as well.

If that isn't what you mean... then just what do you mean?

 

[l0rca]

If you mean by mathematical entities existing that, for instance, if you have 2 of something, and you get 2 more of those things, you'll have 4 of them, then obviously they exist. And I think this is true of all of the more complex cases as well.

If that isn't what you mean... then just what do you mean?

1. object: something that exists as or is perceived as a single separate object
2. philosophy existence: the state of having existence
3. philosophy essential nature of something: the essence or character of something

Encarta ® World English Dictionary © & (P) 1998-2004 Microsoft Corporation. All rights reserved.

 

[andyandy]

I think if these forums were more capable of less redundant debate, I'd be less absolute.
.

you feel like you can overstate your case because you're not intellectually challenged on JREF?

 

[Dr Adequate]

5, 8, and 36 exist, all other mathematical entities are just a comforting fiction.

 

[l0rca]

I just like it here, sometimes.

Certain books I read challenge me intellectually.

A heated debate here wouldn't be intellectual. It would probably be only stubborn, on either my account or another's. When I choose to argue I'm either hoping to learn something, or hoping to share something. If neither is occurring I usually just drop the subject, and mull it over on my own.

 

[Zygar]

Nah, actually you've got it all wrong (and I am a hardcore mathematician, or at least a close approximation thereof). I think you may be confusing the actual laws of physics with our mathematical descriptions of what we think are the laws of physics (check back in about 300 years, and I think you'll find we've had it all wrong---that is to say, by then we'll have come up with mathematical descriptions that better match what we think are the laws of physics).

I think you miss my point. I agree that mathematics and the laws of physics are distinct. I also agree that mathematics stands on its own. But physics itself uses the word "law" only for an axiom that is proven, and the laws of physics require our mathematic entities to be true.

Mathematics is often inspired by real-world objects, but mathematicians *never* works with real-world objects, *always* with abstractions that try to involve the relevant properties of the objects without any of the fiddling details that are unnecessary. This is actually what most people find difficult about mathematics: mathematicians try to abstract the problem to the point that it doesn't actually matter what you apply the results to. This is one of the things that makes it rather difficult to learn, but very powerful to apply.

Take, for instance, the problem of finding the height of a tree from the length of its shadow and the angle of the sun. A pure mathematician only sees the right triangle with the given base and angle---never mind that that triangle does not actually exist (nor, really, can it, other than as a picture in our minds). The mathematician studies the triangle; it matters little to him whether you're applying the trig to find the height of a tree, or the height of the top of a ladder leaning against a wall, or the y-component of a vector given the x-component and the direction, or any number of other 'real-world' applications you can find.

In fact, he's more likely to be fascinated by this idealized triangle and will almost certainly wonder not what he can do with this triangle, but why it works. If he has a philosophical bent, he'll trace the ideas back to the basic assumptions that are being made in coming up with the solution to the problem and begin tinkering with those assumptions to see how changing them changes the final outcome.

Thank you for describing the way my brain works.

It would probably be more accurate to say that the mathematical principles work the way they do because we are trying to describe how the universe operates, not the other way around. If the mathematics does not predict the universe, we change some definitions or assumptions until it does. If, in addition, the model predicts something that we hadn't known before, scientists rush out and test it---if the model does actually predict something and it turns out to be true, it lends the the mystique that the universe is somehow mathematical in nature; if what is predicted does not turn out to be true, we begin tinkering with the model until it agrees with what we observe.

I assert that this is an "if and only if" statement. Mathematics describe the universe if and only if the universe follows mathematics.

In a sense, this mystique is sort of like the mystique some folks have regarding the power of prayer---if they get what they pray for, they believe more strongly in the power of prayer; if they don't, they figure they didn't deserve it in the first place.

I disagree strongly with this. Mathematics are not a religion or a philosophy. They are a science.

Proofs, while the primary tool of mathematicians, are fundamentally dependent on the axioms you start with---that is to say, what assumptions you make. Philosophically, mathematicians like to work with as few assumptions as possible, and prove as much as we can from those assumptions. But change one little thing in any of the basic axioms, and you'll find that any proof that depended on that axiom is no longer valid in the new axiom scheme. In fact, you usually get a new set of proofs and a new set of results (with some commonalities to the old system, if you didn't change too many of the basic assumptions). The mathematics is correct in each of the schemes, but some may be more applicable to the 'real world' than others.

Look up some information on Non-Euclidean geometries if you're curious about this.

Very well written. I agree with your points, but I don't feel that they change the result. Infact, I think you only helped to strengthen my stance.

 

[Iamme]

interesting question.....

my own opinion would be that mathematics is the underlying reality of the universe.

You are in good company. That Huth dude claimed to have solved the age-old riddle as to how the universe's something could have just sprang up from nothing. He claims to have solved it; mathematically. The article was in a Discover Magazine maybe 1-2 years ago.

 

[Blert]

I think you miss my point. I agree that mathematics and the laws of physics are distinct. I also agree that mathematics stands on its own. But physics itself uses the word "law" only for an axiom that is proven, and the laws of physics require our mathematic entities to be true.

I don't see how I'm missing your point---you're basically saying that mathematics exists in the universe independent of our thought; I'm saying that that is simply not true---that mathematics is just our best guess at modeling some observed phenomena. You're saying that mathematics is built in to the universe, I'm saying that it's only built in to our models of the universe (which, if you know anything about the history of science, have turned out invaribly to be wrong, at the very least incomplete and at the very best just an approximation).

I'm saying that mathematics is the tool we use to model the way the universe works; this is a fact, no speculation necessary: we do this. From my point of view, you're making a leap of faith in saying that our tool is actually the way the universe works---that mathematics is the universe's tool, too.

Thank you for describing the way my brain works.

Excellent---if your brain works this way, then you can see the difference between the mathematical object, an idealized triangle (which does not exist, other than in your head) and the actual physical situation. My point here is that the universe exists independent of the way you try to describe it. At best, our idealized triangle merely provides an approximation to the actual fact that exists in the universe.

I assert that this is an "if and only if" statement. Mathematics describe the universe if and only if the universe follows mathematics.

I don't see why this would have to be the case. I'll admit that the congruences are uncanny, but even if we were to assert that we have exactly predicted things in the past, finitely many examples never amount to a proof.

Indeed, one only needs to look at some results in logic that basically amount to the simple truth that any finite axiom system can be either complete (in that everything true can be proven) or consistent, but not both. Since science demands consistency, it follows that no mathematical system can be complete---that the universe always lies beyond the grasp of our tool.

I disagree strongly with this. Mathematics are not a religion or a philosophy. They are a science.

(Umm, I was using an analogy---comparing the mystique surrounding mathematics to the mystique surrounding prayer.)

Very well written. I agree with your points, but I don't feel that they change the result. Infact, I think you only helped to strengthen my stance.

Perhaps some of the above makes the distinction clearer---I must admit, I'm not always a very good communicator.

 

[Zygar]

I don't see how I'm missing your point---you're basically saying that mathematics exists in the universe independent of our thought; I'm saying that that is simply not true---that mathematics is just our best guess at modeling some observed phenomena. You're saying that mathematics is built in to the universe, I'm saying that it's only built in to our models of the universe (which, if you know anything about the history of science, have turned out invaribly to be wrong, at the very least incomplete and at the very best just an approximation).

Newton's Laws have not been changed. They still apply, and are used as the basis for understanding physics. The history of science is certainly filled with examples of science finding a better/more appropriate model of the universe, but the old work has rarely if ever been thrown out the window. As a general rule the new work is based upon the old, and to my knowledge the mathematics used to describe these facts have never been changed simply because the models were wrong.

I'm saying that mathematics is the tool we use to model the way the universe works; this is a fact, no speculation necessary: we do this. From my point of view, you're making a leap of faith in saying that our tool is actually the way the universe works---that mathematics is the universe's tool, too.

Yes, it is a tool. But thinking of it as a tool implies that mathematics are a concrete set of entities which apply to the universe. And we have never had to throw these entities out.

Excellent---if your brain works this way, then you can see the difference between the mathematical object, an idealized triangle (which does not exist, other than in your head) and the actual physical situation. My point here is that the universe exists independent of the way you try to describe it. At best, our idealized triangle merely provides an approximation to the actual fact that exists in the universe.

It is very true that the idealized triangle in one's head is simply an approximation of reality. It causes details that can be important to be lost if one does not attempt to include in the idealized model all of the required features. But an idealized triangle is a mathematical approximation, not a mathematical model of the world. I fail to see how the difference between our internal pictures of the world built through shortcuts of mathematical approximation have anything to do with whether or not mathematical entities are real.

I don't see why this would have to be the case. I'll admit that the congruences are uncanny, but even if we were to assert that we have exactly predicted things in the past, finitely many examples never amount to a proof.

This is our major point of disagreement. This disagreement is essentially the reason for this thread.

Indeed, one only needs to look at some results in logic that basically amount to the simple truth that any finite axiom system can be either complete (in that everything true can be proven) or consistent, but not both. Since science demands consistency, it follows that no mathematical system can be complete---that the universe always lies beyond the grasp of our tool.

Granted. But that means mathematics are incomplete and infinite. This doesn't make the entities any less real.

(Umm, I was using an analogy---comparing the mystique surrounding mathematics to the mystique surrounding prayer.)

Ok. I accept your analogy only because humans do enjoy mystique for some reason.

Perhaps some of the above makes the distinction clearer---I must admit, I'm not always a very good communicator.

Neither am I. But healthy debate only helps.

 

[Soapy Sam]

Is the op not analogous to the famous tree falling in the forest question?
If humanity was eaten by a giant planet eater tomorrow, would pi still exist?
I'd say "no" to both questions.
Noises are artifacts of nervous systems, not trees.
pi is an artifact of minds.

But I agree that the question hinges on the meaning of "real".

 

[gnome]

Another question would be, if there are aliens across the galaxy that have never heard of us, would their mathematics be the same as ours.

I'd say roughly, yes. They might use a different base for their numbers--or a wholly different numbering schema. But I'm sure they still use numbers, logic, set theory. If they perceive shapes, they have geometry. It may be taught in a completely different manner, but I bet most of what they would have is equivalent to ours. I personally think the first possible translations are going to be based on mathematical concepts.

 

[Zygar]

Another question would be, if there are aliens across the galaxy that have never heard of us, would their mathematics be the same as ours.

I'd say roughly, yes. They might use a different base for their numbers--or a wholly different numbering schema. But I'm sure they still use numbers, logic, set theory. If they perceive shapes, they have geometry. It may be taught in a completely different manner, but I bet most of what they would have is equivalent to ours. I personally think the first possible translations are going to be based on mathematical concepts.

I whole-heartedly agree. This is a primary reason for my stance on this subject.

 

[Zygar]

Is the op not analogous to the famous tree falling in the forest question?
If humanity was eaten by a giant planet eater tomorrow, would pi still exist?
I'd say "no" to both questions.
Noises are artifacts of nervous systems, not trees.
pi is an artifact of minds.

But I agree that the question hinges on the meaning of "real".

So ants, squirrels, and deer do not hear? I see this as a very egocentric view of the world. "The world only exists because of humans, so if no humans are around, then it doesn't actually exist."

Having something to comprehend the sound, and pi, is not required for it to be real. It just has to have some impact on the world around it. Even if a tree falling in the forest wasn't percieved by someone, it would still cause vibrations in the trees around it. And pi would still apply to the way rings are formed in the trees.

 

[Soapy Sam]

So ants, squirrels, and deer do not hear? I see this as a very egocentric view of the world. "The world only exists because of humans, so if no humans are around, then it doesn't actually exist."

Having something to comprehend the sound, and pi, is not required for it to be real. It just has to have some impact on the world around it. Even if a tree falling in the forest wasn't percieved by someone, it would still cause vibrations in the trees around it. And pi would still apply to the way rings are formed in the trees.

Last I heard, ants, deer and squirrels all had nervous systems.

The way the zen question is phrased always makes it seem (to me) that it confuses the physics of molecular displacement with the operation of nervous systems. This is an easy error to make as we routinely think of the cause and the resulting sensation as aspects of the same thing. Which they are. But they are not THE SAME thing. The tree falls, shaking the ground and displacing air molecules. But if there is no nervous system there to hear it, there is no noise. The noise is an artifact of brains responding to pressure changes. It's a question of definitions.
(If there was no observer, could there be surprise? I see surprise and noise as equally artifacts of the observer).

I feel we make a similar error when we confuse number with quantity.
"Two" and "Two pianos" are very different things.

Would pi apply to tree rings if we did not know of it? I don't think so.
Would beauty apply to trees in the absence of observers? I don't think so.

Would tree rings be the same shape? The rings in fossil trees are certainly evidence that they would be.