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What Are Numbers?

R

Radical

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I have 10 fingers. The English alphabet has 26 letters. The table has 4 keys. Carbon has 6 protons.

But what's a number? Is it real? If it is real, is it mind-independent or a mental construction? Is it physical/empirical?

I think anyone who gives even the slightest credit to scientists must say numbers are real and mind-independent. How can you not? Numbers are used in every science as a part of the empirical data. If numbers are simply mental constructions, illusions, fictions, or what have you, and do not come from the real world, then science, which is based upon mathematics, also is simply a mental construction, illusion, fiction, or what have you. The understanding of our world is wrapped in mathematics, and it would all have to be thrown out if mathematics was not real.

I think it's also kind of ridiculous to conceive of numbers as empirical. What particle tells me there are 6 of the protons in carbon? There is no "6" particle nor a "2" particle nor a "7,000,000,000" particle. Quantity is not physical.

Quantity is a real, non-physical part of the universe.

Discuss
 
Numbers, like any other part of language, are simply what we use to describe something.
 
Radical said:
I have 10 fingers. The English alphabet has 26 letters. The table has 4 keys. Carbon has 6 protons.

But what's a number? Is it real? If it is real, is it mind-independent or a mental construction? Is it physical/empirical?

I think anyone who gives even the slightest credit to scientists must say numbers are real and mind-independent. How can you not? Numbers are used in every science as a part of the empirical data. If numbers are simply mental constructions, illusions, fictions, or what have you, and do not come from the real world, then science, which is based upon mathematics, also is simply a mental construction, illusion, fiction, or what have you. The understanding of our world is wrapped in mathematics, and it would all have to be thrown out if mathematics was not real.

I think it's also kind of ridiculous to conceive of numbers as empirical. What particle tells me there are 6 of the protons in carbon? There is no "6" particle nor a "2" particle nor a "7,000,000,000" particle. Quantity is not physical.

Quantity is a real, non-physical part of the universe.

Discuss
Click to expand...

It is the same in my area. When you can put it in mathematical form the added insight is wonderful.
 
spud_meister said:
Numbers, like any other part of language, are simply what we use to describe something.
Click to expand...

Are they real? If they're not, they're useless as a description of the real world. You're willing to throw away all understanding of the universe?

the_recruit said:
I know, right? The unreasonable effectiveness of mathematics.
Click to expand...

So they wikipedia page talks about how the author thinks mathematical realism is super awesome (cause it is, minus Platonism), but what is your thoughts? You believe in the existence of immaterial stuffs?
 
This is another example of how our public education system fails us.
 
Radical said:
So they wikipedia page talks about how the author thinks mathematical realism is super awesome (cause it is, minus Platonism), but what is your thoughts? You believe in the existence of immaterial stuffs?
Click to expand...

I find the distinction between material/immaterial badly defined given our profound lack of understanding of the fundamental nature (if there is one) of matter, etc.

I recognize there is a difference between mathematics and, say, chess in that mathematics has a relationship to the reality around us. That the axioms of mathematics are not arbitrary at all, unlike the axioms of chess. Selecting different mathematical axioms would give us a slew of different theorems and formulas that do not correspond to the world around us at all, unlike chess in which changing the rules by which a pawn is allowed to move would be just as 'valid' as any other.

I don't quite know what this means but I have no problem accepting mathematical realism on grounds of indispensability.
 
Radical said:
I have 10 fingers. The English alphabet has 26 letters. The table has 4 keys. Carbon has 6 protons.

But what's a number? Is it real? If it is real, is it mind-independent or a mental construction? Is it physical/empirical?

I think anyone who gives even the slightest credit to scientists must say numbers are real and mind-independent. How can you not? Numbers are used in every science as a part of the empirical data. If numbers are simply mental constructions, illusions, fictions, or what have you, and do not come from the real world, then science, which is based upon mathematics, also is simply a mental construction, illusion, fiction, or what have you. The understanding of our world is wrapped in mathematics, and it would all have to be thrown out if mathematics was not real.

I think it's also kind of ridiculous to conceive of numbers as empirical. What particle tells me there are 6 of the protons in carbon? There is no "6" particle nor a "2" particle nor a "7,000,000,000" particle. Quantity is not physical.

Quantity is a real, non-physical part of the universe.

Discuss
Click to expand...

Set Theory
 
Radical said:
Are they real? If they're not, they're useless as a description of the real world. You're willing to throw away all understanding of the universe?

So they wikipedia page talks about how the author thinks mathematical realism is super awesome (cause it is, minus Platonism), but what is your thoughts? You believe in the existence of immaterial stuffs?
Click to expand...

You are confusing the use of abstract numbers with the reality that they model. The model is not the reality in the same way that a map of the USA is not the USA itself.
 
Radical said:
Ad homenim and didn't provide an alternative. Interesting rhetoric.
Click to expand...

Just a little joke, I apologize.

Others have already hit on the subject. All we are really talking about are abstract objects, something we made up to handle a function of our reality. Much like language, or letters, something we defined as existing in our context to handle basic relation. No matter if we are taking the argument one way or the other as potentially existing outside of our perceptions of reality, what really matters is our use in our reality on our terms. That is an argument about the indispensability of numbers, or letters, or imagery, etc. Once we crossed that point in evolutionary terms then what matters most is the mental exercise of determining realism from abstract objects. Consider all we have advanced on based on these abstract objects.

Use this thought exercise, closed logic conclusion...

1. Number exist because we see then in use.
2. Therefor, because we see them in use with real function we should accept them.
3. Our basis of just about every flavor of Science, Economics, and other academia relies on numbers.
4. Because of 3, we have enough academic suggestion numbers do in fact exist.
5. In conclusion, they must exist because the function of its usage is confirmed.

Just because we cannot pick up a number off a piece of paper and measure its weight does not mean it failed to qualify as reality. We have defined the shape of each number, designed sequencing, designed scope of its use, applied it to our lives in every sense of the word, and designed everything from the most basic of function to handle math in the oldest sense all the way up to complex theoretical math in Quantum Sciences terms we see today.

It is an expanded explanation of the close logic conclusion thought exercise on why numbers must exist, because we made it that way to handle what we needed them to. In Philosophical terms (and for some real fun for this conversation,) we are responsible for their existence and their faults... at the exact same time.
 
csbrown28 said:
Set Theory
Click to expand...

Is set theory a real thing, or is it simply a mental construction?

Also, the axiom of choice, the axiom of infinity, and the axiom of reduciblity are all ad hoc, so the logicist program is a bit iffy. This includes Russell/Whitehead as well as Zermelo/Fraenkel.

William Rea said:
You are confusing the use of abstract numbers with the reality that they model. The model is not the reality in the same way that a map of the USA is not the USA itself.
Click to expand...

Then numbers are not real, and you must conclude there are not 6 protons in Carbon, force does not equal mass times acceleration, and you do not have two legs. That's absolutely ridiculous. You can't honestly conclude that. Everything you do operates upon the supposition that numbers are meaningful. If math is not real, yet meaningful, it would be possible to say "Carbon has 6 protons" and explain why that doesn't mean there are multiple distinct protons within the atom. It can't be done without destroying what we mean by "6".

OrphanSlug said:
Just a little joke, I apologize.

Others have already hit on the subject. All we are really talking about are abstract objects, something we made up to handle a function of our reality. Much like language, or letters, something we defined as existing in our context to handle basic relation. No matter if we are taking the argument one way or the other as potentially existing outside of our perceptions of reality, what really matters is our use in our reality on our terms. That is an argument about the indispensability of numbers, or letters, or imagery, etc. Once we crossed that point in evolutionary terms then what matters most is the mental exercise of determining realism from abstract objects. Consider all we have advanced on based on these abstract objects.

Use this thought exercise, closed logic conclusion...

1. Number exist because we see then in use.
2. Therefor, because we see them in use with real function we should accept them.
3. Our basis of just about every flavor of Science, Economics, and other academia relies on numbers.
4. Because of 3, we have enough academic suggestion numbers do in fact exist.
5. In conclusion, they must exist because the function of its usage is confirmed.

Just because we cannot pick up a number off a piece of paper and measure its weight does not mean it failed to qualify as reality. We have defined the shape of each number, designed sequencing, designed scope of its use, applied it to our lives in every sense of the word, and designed everything from the most basic of function to handle math in the oldest sense all the way up to complex theoretical math in Quantum Sciences terms we see today.

It is an expanded explanation of the close logic conclusion thought exercise on why numbers must exist, because we made it that way to handle what we needed them to. In Philosophical terms (and for some real fun for this conversation,) we are responsible for their existence and their faults... at the exact same time.
Click to expand...

So then math is not real. Numbers don't exist. They're just mental constructions. But why not just say so?

However, you can't bite that bullet; it's suicide. Now there are not 6 protons in Carbon. However, what makes Carbon "Carbon" is the fact it has "6" protons. If it had "8" it would be oxygen. And this isn't simple definitions, the quantity of protons changes everything about it dramatically. Quantity is the most important thing to atomic classification, not because of an arbitrary decision, but because that's the way reality seems to be structured.

So the atoms have a structure that we describe mathematically, but the structure is real, and is not physical. There isn't a particle that is the structure particle. Instead, it is something that an atom has, it is real, and it is not physical. Can we paint the "6" green and shoot the "6" out of a collider? Of course not. We can shoot a proton, but how do we shoot a mathematical structure? The structure obviously exists in the universe as a real thing, and it is just as important as the particle itself. But the structure isn't physical.
 
Radical said:
Is set theory a real thing, or is it simply a mental construction?

Also, the axiom of choice, the axiom of infinity, and the axiom of reduciblity are all ad hoc, so the logicist program is a bit iffy. This includes Russell/Whitehead as well as Zermelo/Fraenkel.
Click to expand...

Does the set of all sets that do not contain themselves contain itself? :mrgreen:
 
Radical said:
Is set theory a real thing, or is it simply a mental construction?

Also, the axiom of choice, the axiom of infinity, and the axiom of reduciblity are all ad hoc, so the logicist program is a bit iffy. This includes Russell/Whitehead as well as Zermelo/Fraenkel.

Then numbers are not real, and you must conclude there are not 6 protons in Carbon, force does not equal mass times acceleration, and you do not have two legs. That's absolutely ridiculous. You can't honestly conclude that. Everything you do operates upon the supposition that numbers are meaningful. If math is not real, yet meaningful, it would be possible to say "Carbon has 6 protons" and explain why that doesn't mean there are multiple distinct protons within the atom. It can't be done without destroying what we mean by "6".

So then math is not real. Numbers don't exist. They're just mental constructions. But why not just say so?

However, you can't bite that bullet; it's suicide. Now there are not 6 protons in Carbon. However, what makes Carbon "Carbon" is the fact it has "6" protons. If it had "8" it would be oxygen. And this isn't simple definitions, the quantity of protons changes everything about it dramatically. Quantity is the most important thing to atomic classification, not because of an arbitrary decision, but because that's the way reality seems to be structured.

So the atoms have a structure that we describe mathematically, but the structure is real, and is not physical. There isn't a particle that is the structure particle. Instead, it is something that an atom has, it is real, and it is not physical. Can we paint the "6" green and shoot the "6" out of a collider? Of course not. We can shoot a proton, but how do we shoot a mathematical structure? The structure obviously exists in the universe as a real thing, and it is just as important as the particle itself. But the structure isn't physical.
Click to expand...

I can't help your reification fallacy of numbers. If you can't get your head around the concept that a map of the USA is not the USA itself then there is little point disappearing up our backsides trying to untie the knots you are putting yourself in.
 
Radical said:
I have 10 fingers. The English alphabet has 26 letters. The table has 4 keys. Carbon has 6 protons.

But what's a number? Is it real? If it is real, is it mind-independent or a mental construction? Is it physical/empirical?

I think anyone who gives even the slightest credit to scientists must say numbers are real and mind-independent. How can you not? Numbers are used in every science as a part of the empirical data. If numbers are simply mental constructions, illusions, fictions, or what have you, and do not come from the real world, then science, which is based upon mathematics, also is simply a mental construction, illusion, fiction, or what have you. The understanding of our world is wrapped in mathematics, and it would all have to be thrown out if mathematics was not real.

I think it's also kind of ridiculous to conceive of numbers as empirical. What particle tells me there are 6 of the protons in carbon? There is no "6" particle nor a "2" particle nor a "7,000,000,000" particle. Quantity is not physical.

Quantity is a real, non-physical part of the universe.

Discuss
Click to expand...

Numbers are just another way of organizing thoughts into patterns as concepts for communicative purposes. What's really going to bend your mind is the latest theory that the universe may not really work with our mathematics... That is to say, "mathematics" is something that WE invented to help us put together and predict things that we do not understand... But are mathematics truly universal?
 
William Rea said:
I can't help your reification fallacy of numbers. If you can't get your head around the concept that a map of the USA is not the USA itself then there is little point disappearing up our backsides trying to untie the knots you are putting yourself in.
Click to expand...

I'm sorry, I directed most of the answer to this objection to OrphanSlug. I should have directed you there.

The map of the united states is an artificial representation of the actual political boarders. But the structure of the map reflects the structure of the actual political boarders. Is the structure real or simply a mental construction? I think that for the case of the US, it's simply a mental construction. However, numbers cannot be dismissed so easily. The mathematical structures are much more fundamental and exist independently of our ability to perceive them/create them.
 
Last edited:
jet57 said:
Numbers are just another way of organizing thoughts into patterns as concepts for communicative purposes. What's really going to bend your mind is the latest theory that the universe may not really work with our mathematics... That is to say, "mathematics" is something that WE invented to help us put together and predict things that we do not understand... But are mathematics truly universal?
Click to expand...

So Carbon does not truly have "6" protons. That's a lie up there with Santa Claus. If numbers are just a means of communicating, we have to ask "is it meaningful or unmeaningful?" If unmeaningful, then everything to do with them is useless. Throw it out. It's worthless. If meaningful, then i ask about what they communicate. They don't communicate a physical thing. They communicate a structure of a physical thing, but the structure itself is not physical. The structure is also real. Therefore, mathematical structures are both real and non-physical.
 
William Rea said:
I can't help your reification fallacy of numbers. If you can't get your head around the concept that a map of the USA is not the USA itself then there is little point disappearing up our backsides trying to untie the knots you are putting yourself in.
Click to expand...

No, you're misunderstanding what OP is getting at. He's not asking about the relationship of a representation to the object which it represents. He's asking about the nature of the object which it represents. I know that what this represents exists:

USA_map-128.png



I know that what this represents does not exist (or, rather, only exists as a concept):

Unicorn_Mount_Patch.png



The question is what does this represent and in what sense does it or does it not exist?

074172-rounded-glossy-black-icon-alphanumeric-number-3.png


You pointing out the relationship between a representation and the object which it represents does not even address the question being asked.
 
Radical said:
So then math is not real. Numbers don't exist. They're just mental constructions. But why not just say so?

However, you can't bite that bullet; it's suicide. Now there are not 6 protons in Carbon. However, what makes Carbon "Carbon" is the fact it has "6" protons. If it had "8" it would be oxygen. And this isn't simple definitions, the quantity of protons changes everything about it dramatically. Quantity is the most important thing to atomic classification, not because of an arbitrary decision, but because that's the way reality seems to be structured.

So the atoms have a structure that we describe mathematically, but the structure is real, and is not physical. There isn't a particle that is the structure particle. Instead, it is something that an atom has, it is real, and it is not physical. Can we paint the "6" green and shoot the "6" out of a collider? Of course not. We can shoot a proton, but how do we shoot a mathematical structure? The structure obviously exists in the universe as a real thing, and it is just as important as the particle itself. But the structure isn't physical.
Click to expand...

Of course it is real, and because it is the applied use of mental constructions (which is why I used the term abstract objects.)

Speak of, the number of protons in any element is equal to the atomic number. So Carbon has 6 protons. In this case Neutrons is then equal to the atomic weight minus the atomic number, or 12-6=6. Electrons is the same as the atomic number, or 6.

That is not arbitrary but an extension of our application of math (numbers.) And they are very real for not just the mental exercise I've put forth, but because you see the application of that usage in terms of this field of science. Atomic weight is then most important thing to classification of elements, and you already know why.

So how could this not be real?
 
Radical said:
Is set theory a real thing, or is it simply a mental construction?
Click to expand...

It sounds to me like you need to start from the beginning.

Do we exist?

One could argue that the fact that I can ask that question lends some credence to my claim that I, do in fact, exist.

Do my senses tell me anything about reality as it actually is?

Unfortunately, the only reality is that there is no way to prove that my senses are giving me accurate information about reality because I'm bound by those same senses to figure it out. I make the assumption that my senses are in fact sometimes accurate.

From there everything else that I know to be true should be able to be shown to be true via evidence.

Set theory lends evidence that numbers are something useful in that they are the same for everyone and can be used to do some pretty cool s**t. But if you are asking me if numbers transcend the human mind, that is they exist in all places at all times I would say the answer is no. I can imagine a universe of fluid where nothing is solid and no sets can be arranged.

So the idea that sets do in fact exist in this universe can be shown to be true via evidence, but I don't think that numbers transcend this universe to all other universes.

Or have I terribly over thought your question?
 
Radical said:
I think anyone who gives even the slightest credit to scientists must say numbers are real and mind-independent. How can you not? Numbers are used in every science as a part of the empirical data. If numbers are simply mental constructions, illusions, fictions, or what have you, and do not come from the real world, then science, which is based upon mathematics, also is simply a mental construction, illusion, fiction, or what have you.
Click to expand...

Something doesn't follow, here. I think it's at the very beginning, when you say "anyone who gives even the slightest credit to scientists..." Look, suppose nominalism with respect to numbers is correct, and suppose this has the implication that any scientific theory which makes use of numbers is also a name or bundle of names. How does this mean that scientists don't deserve credit, or that the names in question aren't correct in every sense?

You seem to simply dismiss nominalism without actually dealing with the arguments in its favor (I am not, incidentally, a nominalist).

Radical said:
The understanding of our world is wrapped in mathematics, and it would all have to be thrown out if mathematics was not real.
Click to expand...

Also doesn't follow. Why throw out what works, even if the reason it works has nothing to do with reality just as such?

Radical said:
I think it's also kind of ridiculous to conceive of numbers as empirical. What particle tells me there are 6 of the protons in carbon? There is no "6" particle nor a "2" particle nor a "7,000,000,000" particle.
Click to expand...

Finally, what is this? You go from talk of numbers being empirical to number "particles." There's no obvious relationship there.
 
the_recruit said:
No, you're misunderstanding what OP is getting at. He's not asking about the relationship of a representation to the object which it represents. He's asking about the nature of the object which it represents. I know that what this represents exists:
...
You pointing out the relationship between a representation and the object which it represents does not even address the question being asked.
Click to expand...

I disagree, the example given in the OP specifically conflates the reality of numbers with the reality it is used to describe.

Radical said:
...I think anyone who gives even the slightest credit to scientists must say numbers are real and mind-independent. How can you not? Numbers are used in every science as a part of the empirical data. If numbers are simply mental constructions, illusions, fictions, or what have you, and do not come from the real world, then science, which is based upon mathematics, also is simply a mental construction, illusion, fiction, or what have you. The understanding of our world is wrapped in mathematics, and it would all have to be thrown out if mathematics was not real....
Click to expand...

1. Numbers are not empirical data, they are a description of empirical data.

2. Numbers are not derived from the real world, they are a description of the real world.

3. Science uses mathematics to model/approximate/describe phenomena in the real world. The model/approximate/describe is NOT the real world and does not have any consequence for the real world which carries on doing what it does regardless of our attempts to model/approximate/describe it.

4. Our understanding of the world is achieved via mathematics, the world does not operate as a consequence of mathematics. F=mg is a way to model/approximate/describe falling objects, they do not fall because F=mg just like if I were to draw a map of a country it would not spring into existence because of that.

5. Mathematics can be done in an abstract sense without ascribing any meaning to the numbers or symbols.

6. There are six protons in a Carbon atom because that is how physical reality is, mathematics does not change that, it helps us understand that.

7. There is a conflation of number, quantity,mathematics etc going on all over this thread so, I'm out of it. that way lies insanity.
 
the_recruit said:
No, you're misunderstanding what OP is getting at. He's not asking about the relationship of a representation to the object which it represents. He's asking about the nature of the object which it represents. I know that what this represents exists:

USA_map-128.png



I know that what this represents does not exist (or, rather, only exists as a concept):

Unicorn_Mount_Patch.png



The question is what does this represent and in what sense does it or does it not exist?

074172-rounded-glossy-black-icon-alphanumeric-number-3.png


You pointing out the relationship between a representation and the object which it represents does not even address the question being asked.
Click to expand...

You're my new best friend. I wish I could like this post a thousand times over.
 
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