Do numbers EXIST? - Numberphile
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When you say that a mathematical fictionalist (MF) does not believe in numbers I think you mean that they do not believe in things like the number two existing as their own abstract object outside of time and space, but that MF's DO believe that they exist as symbols in a formal system, and/or as a potential mental state in one or more minds. Is that a fair representation of the MF's view?
Also, I would be interested in hearing you expound on the statement that MF's do not believe that -mathematical discourse is true-. If I believed that within the formal rules of maths that 2+2=4 and I believed that this formal system happened to be very useful at predicting behavior in time and space, can I still be a MF? What does -True- mean in your statement? Does it strictly refer to whether the abstract objects exist?
Myself, I think that if there were suddenly no human minds anywhere in the universe there would also be no 'Twoness'. Does that make me an MF instead of a nominalist?
I concede that even if there were no minds in the universe a binary star would still consist of two stars, but the 'twoness' of that statement only comes from my ability to imagine a mind being there to make that statement. If I substitute 'anger' for 'twoness' I think people would agree more readily.
I apologies that I do not have a very developed vocabulary for this discussion. Thanks for any clarity you can bring to my questions.
Date: 2022-04-08
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Comments and reviews: 9
Lou
When I teach high school mathematics I start by putting four statements on a blackboard:
1. The speed of light in a vacuum is 3-10-8 m/s.
2. Christopher Columbus discovered America.
3. Romeo and Juliet are in love.
4. 2 + 3 = 5.
I claim that these are all true.
1. In physics, you need something that is close enough to be useful. 3-10-8 is close, but maybe not close enough for some calculations, in which case, use a better approximation. No number is exact, but if you're close enough physics doesn't care, so call it true.
2. Not exactly. He certainly didn't know it was there, but neither did anyone else in Europe. In any case, he opened up the Americas for better or worse to the Europeans.
3. Sure, except Romeo and Juliet don't exist. They're fictional characters.
Statement 1 is true in the context of descriptions of the physical world, which are always necessarily approximate. Truly exact measurements are Heisenbergially ambiguous, and therefore approximate.
Statement 2 is (only) true in a historical context, and from the point of view of the late Renaissance Europeans.
Statement 3 is entirely dependent on poetic license, which is itself another definition of truth.
Statements 1, 2, and 3 can be considered -true- in the contexts in which they are stated. It is in the context of mathematics that statement 4 is true. That is, given the common definitions the symbols involved it's a true statement. What I tell my students is that mathematics is precisely every statement that is either true or false in the same way as statement 4.
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When I teach high school mathematics I start by putting four statements on a blackboard:
1. The speed of light in a vacuum is 3-10-8 m/s.
2. Christopher Columbus discovered America.
3. Romeo and Juliet are in love.
4. 2 + 3 = 5.
I claim that these are all true.
1. In physics, you need something that is close enough to be useful. 3-10-8 is close, but maybe not close enough for some calculations, in which case, use a better approximation. No number is exact, but if you're close enough physics doesn't care, so call it true.
2. Not exactly. He certainly didn't know it was there, but neither did anyone else in Europe. In any case, he opened up the Americas for better or worse to the Europeans.
3. Sure, except Romeo and Juliet don't exist. They're fictional characters.
Statement 1 is true in the context of descriptions of the physical world, which are always necessarily approximate. Truly exact measurements are Heisenbergially ambiguous, and therefore approximate.
Statement 2 is (only) true in a historical context, and from the point of view of the late Renaissance Europeans.
Statement 3 is entirely dependent on poetic license, which is itself another definition of truth.
Statements 1, 2, and 3 can be considered -true- in the contexts in which they are stated. It is in the context of mathematics that statement 4 is true. That is, given the common definitions the symbols involved it's a true statement. What I tell my students is that mathematics is precisely every statement that is either true or false in the same way as statement 4.
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Luck
I used to be a nominalist before I entered university. Then linear algebra started, and this is when nominalism totally collapsed for me. It was a momentary aha moment. All it took was to reflex on the axiomatic definition of what a vector is. Vectors aren't unique concepts in this manner, but it was the thing that striked me. Vectors and vector space create a world completely detached from reality. You don't need anything real to define vector space, and you actually can't use anything real to define a vector. Some vectors can be discribed using numbers and some basis, but vectors aren't those sequences of numbers. And there are vectors that can't be fully described using numbers, for example vectors in spaces with infinite dimensionality. And everything can be a vector if operations are defined. Vectors are literally -anything, for which operations of addition and scaling is defined and well behaved-. Look, I say, that I define operation multiply(my bed, any number) = my bed; add(my bed, my bed) = my bed, vector space = set, containing only my bed. Now my bed (a literal physical object that I am sitting on right now) is a vector, because it satisfies all the axioms of vector space! Not a description of the bed, not a representation of the bed, but the bed itself is a vector. And no, it's not like some vector is used to represent my bed, but my bed IS a vector in this context.
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I used to be a nominalist before I entered university. Then linear algebra started, and this is when nominalism totally collapsed for me. It was a momentary aha moment. All it took was to reflex on the axiomatic definition of what a vector is. Vectors aren't unique concepts in this manner, but it was the thing that striked me. Vectors and vector space create a world completely detached from reality. You don't need anything real to define vector space, and you actually can't use anything real to define a vector. Some vectors can be discribed using numbers and some basis, but vectors aren't those sequences of numbers. And there are vectors that can't be fully described using numbers, for example vectors in spaces with infinite dimensionality. And everything can be a vector if operations are defined. Vectors are literally -anything, for which operations of addition and scaling is defined and well behaved-. Look, I say, that I define operation multiply(my bed, any number) = my bed; add(my bed, my bed) = my bed, vector space = set, containing only my bed. Now my bed (a literal physical object that I am sitting on right now) is a vector, because it satisfies all the axioms of vector space! Not a description of the bed, not a representation of the bed, but the bed itself is a vector. And no, it's not like some vector is used to represent my bed, but my bed IS a vector in this context.
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I'mnotveryfondofit
I don't know with certainty but I think a pragmatists (or even Wittgenstein) response to a Platonist on numbers would be that their existence is a concern that is falsely deemed relevant by the Platonists idealistic assumptions. The Platonist, thought they are unaware, applies similar ontological qualifiers to numbers as they do objects which are bound by space and time with slight differences. This is why the Platonist has to construct the existence of a universal realm (Platonic ideals) so as to reasonably make sense of the idea of number and its having ontological status. The pragmatists and philosophers of language following Wittgenstein (possibly even a Kantian) would do away with Platonic excesses and say that number is (something like a) convenient and totally reliable mental construct. Numbers do not exist in the sense that Plato suggests. The form (following Platonic language) number is a concept created by humans to describe particulars (again with Plato) which categorically fall under that form. Similar idea: soup can as a form does not exist in some universal, extraterrestrial or absolute realm. Such ideas are the left-over biases of certain religious worldviews and cultural influences. The particular soup can pre exists the form. Soup can is a convenient designator applied to empirical phenomenon which the brain then categorizes.
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I don't know with certainty but I think a pragmatists (or even Wittgenstein) response to a Platonist on numbers would be that their existence is a concern that is falsely deemed relevant by the Platonists idealistic assumptions. The Platonist, thought they are unaware, applies similar ontological qualifiers to numbers as they do objects which are bound by space and time with slight differences. This is why the Platonist has to construct the existence of a universal realm (Platonic ideals) so as to reasonably make sense of the idea of number and its having ontological status. The pragmatists and philosophers of language following Wittgenstein (possibly even a Kantian) would do away with Platonic excesses and say that number is (something like a) convenient and totally reliable mental construct. Numbers do not exist in the sense that Plato suggests. The form (following Platonic language) number is a concept created by humans to describe particulars (again with Plato) which categorically fall under that form. Similar idea: soup can as a form does not exist in some universal, extraterrestrial or absolute realm. Such ideas are the left-over biases of certain religious worldviews and cultural influences. The particular soup can pre exists the form. Soup can is a convenient designator applied to empirical phenomenon which the brain then categorizes.
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jsclipse
Do we not substitute numbers for our view of reality? I always thought I did. If I calculate one billion dollars, does the math support the number of representative one dollar bills? I have never seen one billion individual one dollar bills in one place to be counted. Do they exist? Must they? I have never seen the actual dust particles in a space nebula. Can they be calculated? Do they exist? Must they? I have never seen a second or a minute - only their representations on a clock. Does time exist? Would numeric substitution (associating numbers with things) easily apply to abstract items too? Is time real or abstract? (Time is an illusion - lunch time doubly so. -- Douglas Adams. Why count it? If (whatever) existed their number would be (answer? If an irrational number is the product, then the substituted item is irrational? No. Using a calculation of PI can assist with creating a circular patio: A circular patio with a 5 foot radius (5-5) X 3. 1415926535 = an area of 78. 53 on paper but the area can be measured once created without this formula. Irrational numbers can be used to create the physical and can be used to associate a reality interdependently as easily as one stick plus another stick equals two sticks. [One, two, buckle my shoe; three, four, shut the door; five, six, pick up sticks. ] What if there are no shoes? no doors? no sticks?
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Do we not substitute numbers for our view of reality? I always thought I did. If I calculate one billion dollars, does the math support the number of representative one dollar bills? I have never seen one billion individual one dollar bills in one place to be counted. Do they exist? Must they? I have never seen the actual dust particles in a space nebula. Can they be calculated? Do they exist? Must they? I have never seen a second or a minute - only their representations on a clock. Does time exist? Would numeric substitution (associating numbers with things) easily apply to abstract items too? Is time real or abstract? (Time is an illusion - lunch time doubly so. -- Douglas Adams. Why count it? If (whatever) existed their number would be (answer? If an irrational number is the product, then the substituted item is irrational? No. Using a calculation of PI can assist with creating a circular patio: A circular patio with a 5 foot radius (5-5) X 3. 1415926535 = an area of 78. 53 on paper but the area can be measured once created without this formula. Irrational numbers can be used to create the physical and can be used to associate a reality interdependently as easily as one stick plus another stick equals two sticks. [One, two, buckle my shoe; three, four, shut the door; five, six, pick up sticks. ] What if there are no shoes? no doors? no sticks?
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TexasFriedCriminal
Nominalism runs into much more fundamental problems than infinite and transcendent numbers.
One thing is that reality underdetermines quantities. If there are two pebbles on a plate, are there two objects (pebble one, pebble two) or three (pebble one, pebble two, both pebbles) or one (the pebbles) and what about the plate? Nominalism inherits all the problems concerning the identity of objects as only when identity is fixed, the reference of -one- can be fixed. And that, I think cannot be done. Hence, there are no actual numbers in the world.
Similarly, where in the world is addition? How do I know that the pebbles on the plate are a case of 1+1 and not of 0+2 or 2+0? This question is a issue for all algebraic operations.
Finally, nominalism requires you reject literalism about mathematical statements. But if you do that, you need to provide consistent and complete rules of translation to go from what we say to what it actually means. And those may be pretty hard to come by.
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Nominalism runs into much more fundamental problems than infinite and transcendent numbers.
One thing is that reality underdetermines quantities. If there are two pebbles on a plate, are there two objects (pebble one, pebble two) or three (pebble one, pebble two, both pebbles) or one (the pebbles) and what about the plate? Nominalism inherits all the problems concerning the identity of objects as only when identity is fixed, the reference of -one- can be fixed. And that, I think cannot be done. Hence, there are no actual numbers in the world.
Similarly, where in the world is addition? How do I know that the pebbles on the plate are a case of 1+1 and not of 0+2 or 2+0? This question is a issue for all algebraic operations.
Finally, nominalism requires you reject literalism about mathematical statements. But if you do that, you need to provide consistent and complete rules of translation to go from what we say to what it actually means. And those may be pretty hard to come by.
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Norman
There's a problem with pointing out all the successes of science and engineering that are based on mathematics, and that's all the failures of mathematical theories. If maths is real, and the world is mathematical, then how is it possible for a mathematical theory to be wrong? Take Newton's theory of gravity as an example. Physicists in the 18th and 19th centuries might have said that the success of Newton's theory of gravity demonstrates that the 1/r--2 force law of gravity is real and true. But then Einstein came along and gave a totally different (but also mathematical) theory of gravity, which (apart from its theoretical advantages) more accurately reflects observations. So where does this leave Newton's theory? A common sense assessment would conclude that it was a human error, there never was a 1/r--2 force of gravity; we invented it and imposed it on the world and we've now realised that it doesn't actually fit. Is all of science and engineering like this?
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There's a problem with pointing out all the successes of science and engineering that are based on mathematics, and that's all the failures of mathematical theories. If maths is real, and the world is mathematical, then how is it possible for a mathematical theory to be wrong? Take Newton's theory of gravity as an example. Physicists in the 18th and 19th centuries might have said that the success of Newton's theory of gravity demonstrates that the 1/r--2 force law of gravity is real and true. But then Einstein came along and gave a totally different (but also mathematical) theory of gravity, which (apart from its theoretical advantages) more accurately reflects observations. So where does this leave Newton's theory? A common sense assessment would conclude that it was a human error, there never was a 1/r--2 force of gravity; we invented it and imposed it on the world and we've now realised that it doesn't actually fit. Is all of science and engineering like this?
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Sonnenhafen
if you begin as a nominalist and think about the physical structures in the world, you learn the notion of a number, you learn to use it and slowly build up a mapping between physical world and maths symbols, you begin to think about these mathematical structures as objects naturally because thats what our brain is used to, it refers to repeating structures that are constructed after rules as their own entities, therefore lifting structures to thoughts and therefore to objects of our recognition. On this stage there is only a difference in interpretation of the information, physical objects and -thought--objects are really the same for our inner recognition, meaning that they're both entities that our mind can refer to and therefore the notion of existence and objectification becomes the same thing, what eventually constructs the things we mean when we think of numbers and maths as objects.
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if you begin as a nominalist and think about the physical structures in the world, you learn the notion of a number, you learn to use it and slowly build up a mapping between physical world and maths symbols, you begin to think about these mathematical structures as objects naturally because thats what our brain is used to, it refers to repeating structures that are constructed after rules as their own entities, therefore lifting structures to thoughts and therefore to objects of our recognition. On this stage there is only a difference in interpretation of the information, physical objects and -thought--objects are really the same for our inner recognition, meaning that they're both entities that our mind can refer to and therefore the notion of existence and objectification becomes the same thing, what eventually constructs the things we mean when we think of numbers and maths as objects.
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greyfade
Platonism is the theory of the mentally infirm.
Nominalism is the theory of the cowardly Platonist.
Fictionalism is the theory of those who fear philosophy and just want to escape the quandary.
The truth, which I might term -positism, - (the link to metaphilosophy is accidental, but happily coincident) is that numbers do not exist _per_ _se_, rather that numbers are the production of a set of constructive rules which coincide with their inherent usefulness, but representative only of a state.
Consider the chessboard, on which pieces are arranged in a non-initial state, deep into a complex game. Does this board position exist? The platonist says yes, and is wrong. The nominalist says yes, and is also wrong. The fictionalist says no, and is not _even_ wrong. The positist says that the state is the product of an interesting game that can be studied and learned from.
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Platonism is the theory of the mentally infirm.
Nominalism is the theory of the cowardly Platonist.
Fictionalism is the theory of those who fear philosophy and just want to escape the quandary.
The truth, which I might term -positism, - (the link to metaphilosophy is accidental, but happily coincident) is that numbers do not exist _per_ _se_, rather that numbers are the production of a set of constructive rules which coincide with their inherent usefulness, but representative only of a state.
Consider the chessboard, on which pieces are arranged in a non-initial state, deep into a complex game. Does this board position exist? The platonist says yes, and is wrong. The nominalist says yes, and is also wrong. The fictionalist says no, and is not _even_ wrong. The positist says that the state is the product of an interesting game that can be studied and learned from.
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SZP
The objections to the mathematical platonism leads him into a corner, that is a beautiful corner he doesnt want to go down, that is Truth is grounded in something outside of our minds. Aristotle proved this with his transcendental argument of the law of non contradiction, where if you were to try and deny the law, you would be invoking the law. Maths is a language of logic. Try to deny the number 1 exists and you would be invoking, 1, which necesitates 2, 3, 4. N.
And secondly, any claim you make on the nature of maths is a Truth claim, which means saying maths isnt True or that numbers are just practical is a Truth claim, maths has inbuilt Truth values in it, that is you OUGHT to follow what is true. Any direction you go you will be entering into a self destructive contradiction, that is all philosophies but One.
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The objections to the mathematical platonism leads him into a corner, that is a beautiful corner he doesnt want to go down, that is Truth is grounded in something outside of our minds. Aristotle proved this with his transcendental argument of the law of non contradiction, where if you were to try and deny the law, you would be invoking the law. Maths is a language of logic. Try to deny the number 1 exists and you would be invoking, 1, which necesitates 2, 3, 4. N.
And secondly, any claim you make on the nature of maths is a Truth claim, which means saying maths isnt True or that numbers are just practical is a Truth claim, maths has inbuilt Truth values in it, that is you OUGHT to follow what is true. Any direction you go you will be entering into a self destructive contradiction, that is all philosophies but One.
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