Actual philosophy
Time and time again I find myself coming back to this question: What is a number? A miserable little pile of secrets.
First I think we should ask, where do numbers (numbers in general, not particular numbers) come from? Do we discover them, or do we invent them? If we discover numbers, that implies that numbers are an intrinsic part of the universe and may even transcend it. This is the stance of platonism. Platonism has by now been pretty much debunked in all areas of philosophy expect one: the philosophy of mathematics. A hardcode platonist would even argue that there is some place where pi and 5 are as real there as the screen in front of you is real here.
While that idea has a certain aesthetic appeal, it's not without its problems. For example, the string "2 + 2 = 4" has absolutely no meaning outside of a specific mathematical framework. You can't prove that statement using Euclidean geometry. Logicism states that -- since numbers are simply consequences of symbolic manipulations that we have defined as valid (e.g. if "x" is a well-formed statement, then so is "not x") being performed on basic sentences (e.g. A is A) -- numbers exist only because we do.
While logicism is an excellent argument, I think it fails to explain certain important facts. When we first dabble with mathematics, it's never in formal terms. Imagine trying to teach young children arithmetic using Peano's axioms, or ZFC set theory. We only come up with formalities once we've mastered the particulars and we want to extrapolate from what we know to explore new territories. We're booting our brains, if you will. In this sense, our initial understanding of mathematics would appear to be either intuitive or empirical, both of which suggest that numbers are a priori to our understanding of them.
What do you think?
First I think we should ask, where do numbers (numbers in general, not particular numbers) come from? Do we discover them, or do we invent them? If we discover numbers, that implies that numbers are an intrinsic part of the universe and may even transcend it. This is the stance of platonism. Platonism has by now been pretty much debunked in all areas of philosophy expect one: the philosophy of mathematics. A hardcode platonist would even argue that there is some place where pi and 5 are as real there as the screen in front of you is real here.
While that idea has a certain aesthetic appeal, it's not without its problems. For example, the string "2 + 2 = 4" has absolutely no meaning outside of a specific mathematical framework. You can't prove that statement using Euclidean geometry. Logicism states that -- since numbers are simply consequences of symbolic manipulations that we have defined as valid (e.g. if "x" is a well-formed statement, then so is "not x") being performed on basic sentences (e.g. A is A) -- numbers exist only because we do.
While logicism is an excellent argument, I think it fails to explain certain important facts. When we first dabble with mathematics, it's never in formal terms. Imagine trying to teach young children arithmetic using Peano's axioms, or ZFC set theory. We only come up with formalities once we've mastered the particulars and we want to extrapolate from what we know to explore new territories. We're booting our brains, if you will. In this sense, our initial understanding of mathematics would appear to be either intuitive or empirical, both of which suggest that numbers are a priori to our understanding of them.
What do you think?
Numbers are an abstraction of quantity/value/etc. and are a part of language.
Numbers are not physically "real", we just count/measure stuff and then we use numbers to express ourselves.
If I misunderstood the topic, I apologize.
Numbers are not physically "real", we just count/measure stuff and then we use numbers to express ourselves.
If I misunderstood the topic, I apologize.
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Tell me if I'm wrong here, I've always thought that I just don't have the mind for philosophy. But the problem is that the term "numbers" is just too broad to define in specific terms.
We have integers which are simply an extension of our language. These are the Arabic symbols that we use as short hand and are no different from the shorthand symbols "#, %, &, etc.". These are not real, it is simply faster to type and\or process them compared to if we spelled out the word that they stood for. (IMHO) The major problem with teaching\learning mathematics today is that this system was only ever "good enough" and today it has become completely inadequate to represent the complex ideas that we are trying to communicate with each other.
Then we have ratios, formulas and functions. These ARE real and can be proven and demonstrated consistently in any language or numerical base that you choose. Remember that pi\phi are not numbers, they are ratios that consistently reduce to the same numbers. This is why they are seen time and time again.
We have integers which are simply an extension of our language. These are the Arabic symbols that we use as short hand and are no different from the shorthand symbols "#, %, &, etc.". These are not real, it is simply faster to type and\or process them compared to if we spelled out the word that they stood for. (IMHO) The major problem with teaching\learning mathematics today is that this system was only ever "good enough" and today it has become completely inadequate to represent the complex ideas that we are trying to communicate with each other.
Then we have ratios, formulas and functions. These ARE real and can be proven and demonstrated consistently in any language or numerical base that you choose. Remember that pi\phi are not numbers, they are ratios that consistently reduce to the same numbers. This is why they are seen time and time again.
I remember recently watching a minutephysics video on YouTube , featuring the author of the then released book on a similar subject.
The video was titled "Is Our Universe Purely Mathematical".You should watch it.
Going by the idea of the video , numbers are inct real , and our universe can actually be explained by considering just Math.For example consider an electron it can be represented as the numbers -1 , +- 1/2 and some other "Quantum numbers" , it is these numbers which corresponds to the intrinsic properties of the electron , so it being a fermion has a fractional spin , it has a negative charge etc.So you can say that these numbers represent the fundamental particles , while our macro world is a manifestation of these particles.
The video was titled "Is Our Universe Purely Mathematical".You should watch it.
Going by the idea of the video , numbers are inct real , and our universe can actually be explained by considering just Math.For example consider an electron it can be represented as the numbers -1 , +- 1/2 and some other "Quantum numbers" , it is these numbers which corresponds to the intrinsic properties of the electron , so it being a fermion has a fractional spin , it has a negative charge etc.So you can say that these numbers represent the fundamental particles , while our macro world is a manifestation of these particles.
I wouldn't say that quantum numbers represent particles. Describe their states, within the bounds of a particular mathematical model, maybe.
| Numbers are an abstraction of quantity/value/etc. and are a part of language. Numbers are not physically "real", we just count/measure stuff and then we use numbers to express ourselves. |
That aside, the question is not whether numbers are physically real like electrons are -- as they are obviously not -- but in what sense do they exist. What is their ontological status?
| We have integers which are simply an extension of our language. These are the Arabic symbols that we use as short hand and are no different from the shorthand symbols "#, %, &, etc.". |
| The major problem with teaching\learning mathematics today is that this system was only ever "good enough" and today it has become completely inadequate to represent the complex ideas that we are trying to communicate with each other. |
| Remember that pi\phi are not numbers, they are ratios that consistently reduce to the same numbers. |
Maybe you could say that number systems are languages which allow us to precisely describe relationships between things.
| For example? (Regarding the inadequacy of our numbering system) |
Irrational numbers for one. They only go on forever because the current system we use does not have anything in place to take into account scale. We know that at some point dividing something again means that the data set ceases to represent what it originally did but there is no set way to express that in out current system. Say you have a circle made of wood, the physical boundaries exist and are observable as the edges of that circle. You want the ratio of the circumference to the diameter so you start dividing units of distance that you measured. At some point the units of distance you are working with have to cut off since both the radius and the diameter are measurable finite values. But our current system has no way of saying what that point is. EDIT: This gif shows what I mean perfectly: http://upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Pi-unrolled-720.gif/220px-Pi-unrolled-720.gif
You can do this with a circle and measure the resulting value, but you can't calculate it because our numbering system is not up to the task.
I felt that I had a stronger argument to this point using imaginary\complex numbers but as I typed it out I realized that I was just stuck on an abstraction. I may have defeated my own argument here.
| A ratio is a number. The string "the ratio of a circle's diameter to its circumference" is indeed not a number, but the ratio of a circle's diameter to its circumference is a number, which is pi. |
Yes, I agree with you here. My point was more to the effect that a ratio is normally not a set value, the fact that they always resolve to the same values with circles and sphere's is anecdotal to those instances of a ratio. We plug them in because we know what they will be ahead of time for those particular use cases. I wanted to point this out so that we don't point to either one and say that these integers represent real world values, they are just ratios that happen to resolve to the same value every time.
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I would say that... numbers are the end result of a field. This field has a few specific rules:
-It is one-dimensional
-All points on this field are unique (no two points overlap that cannot be simplified into the same point (3 does not equal 5)
-There is a metric function for this field
-This metric function can only measure the distance between two points, due to the one-dimensionality of the field
-No two same points have different distances between each other
-Distance is not unique- there is no distance between any two points that is not held by another set of two points
-The distance between any two points is a scaler value that, in itself, is equal to a point in this field
Numbers, in other words, are a mathematical construction. Everything else that comes along with numbers is pure coincidence.
-It is one-dimensional
-All points on this field are unique (no two points overlap that cannot be simplified into the same point (3 does not equal 5)
-There is a metric function for this field
-This metric function can only measure the distance between two points, due to the one-dimensionality of the field
-No two same points have different distances between each other
-Distance is not unique- there is no distance between any two points that is not held by another set of two points
-The distance between any two points is a scaler value that, in itself, is equal to a point in this field
Numbers, in other words, are a mathematical construction. Everything else that comes along with numbers is pure coincidence.
| You can do this with a circle and measure the resulting value, but you can't calculate it because our numbering system is not up to the task. |
You can't measure the resulting value perfectly either... no matter how precise your measuring instrument there will always be uncertainty so the fact that you cannot calculate pi to a finite precision isn't surprising.
I don't really think that this shows that the numbering system is inadequate.
@Ispil
where do complex numbers fit in this field then?
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@ Void Life: Are you being purposefully pedantic here? There is a certain point where further precision in measurement adds nothing of value to an equation. Even if that was not the case, what stops someone from using an electron microscope to measure the remainder of the line?
@ Void Life
Simple- it's a different field. Make it two dimensional, where the distance between any two points becomes a touch more complicated depending on how you represent any two points (cis, re^it, a+bi). Otherwise, it's the same system. Just apply some trigonometry to the distance equations, and the result is the same.
Simple- it's a different field. Make it two dimensional, where the distance between any two points becomes a touch more complicated depending on how you represent any two points (cis, re^it, a+bi). Otherwise, it's the same system. Just apply some trigonometry to the distance equations, and the result is the same.
| Numbers, in other words, are a mathematical construction. Everything else that comes along with numbers is pure coincidence. |
Ispil wrote:
and helios replied:
I agree with helios here. The abstract mathematics of number systems (Peano's postulates, the rationals, the reals, etc.) has its genesis in the human mind of something that is perceived to be part of nature, not so much as an object that exists, but rather as a property or attribute of certain configurations. For example, if there is a box containing five apples, then "fiveness" is a property of the box and the apples taken collectively. In a primitive sense, how would we quantify the apples in the box? By "counting" them with our fingers. We match each apple with one of our fingers. If we run out of apples before we run out of fingers there are less apples then fingers and if we run out of fingers before apples then there are more apples than fingers. If we run out of apples at the same time that we run out of fingers then there are the same number of apples and fingers. This is exactly what occurs mathematically in comparing the cardinality of two sets. Two sets have the same number of elements if there exists a one-to-one onto function (a bijection) that maps one set on to the other. Eventually we give names to these quantities that we intuited (one, two, three, four, etc.). It seems to me that the practitioners of a language would need to have developed well above the level of a small primitive tribe before they would invent words for very large numbers such as million or billion. Even today we abandon the process of inventing new words for very large numbers and use scientific notation with powers of ten giving us the ability to represent arbitrarily large numbers, but when we do so we lose any intuitive understanding of the quantity that such numbers represent. For example, no one has any conception of how 1010000 and 1010001 differ in quantity since there is nothing in nature to compare them with. Though these two numbers can be represented, we are forced to revert to primitive notions of cardinality and the most that we can say is that the second number is bigger than the first. (You need more fingers to count the second than the first. A lot more.)
If quantity (and hence numbers) is something in nature that can be perceived by the human mind, is that capability unique to humans? I think not. Consider a wolf (not a domesticated animal to avoid influence by man) with a liter of five pups. She moves a short distance away and her pups naturally follow but one of them straggles. She suddenly recognizes that one is missing so she backtracks, finds the straggler, resumes her move with all five now in tow. Clearly to recognize that one is missing she must in some sense be counting. Even if one attempts to evade the argument by saying she's not really counting but perhaps had stored in her brain a collection of the scents of her five pups and simple matches those stored scents with the four pups that are present and thus discovers that one is missing, but she doesn't really have any conception of the idea of "fiveness" so she's not really counting. But if she can recognize that one of the five stored scents is not matched up she must in some manner be engaging in some type of cardinality comparison, primitive as it may be. I conclude therefore, that numbers or quantity are inherent and inseparable from nature and are not something created by the human mind but something that is perceived by mind.
helios: [of numbers]
I believe that is a question that cannot be answered within the framework of naturalism and the metaphysical assumptions of modern science. For me, as a Christian, the answer to the question as to why the natural world and the Universe are the way they are, as to why the human mind can perceive such order and structure in the Universe as revealed by science, can only be found in the Creator and beyond that it is impossible to push.
| Numbers, in other words, are a mathematical construction. Everything else that comes along with numbers is pure coincidence. |
and helios replied:
| This is unsatisfactory. It doesn't address at all why mathematics appears to occur naturally in the universe. If it's a purely human invention, it makes more sense that it'd be inapplicable to the natural world. |
I agree with helios here. The abstract mathematics of number systems (Peano's postulates, the rationals, the reals, etc.) has its genesis in the human mind of something that is perceived to be part of nature, not so much as an object that exists, but rather as a property or attribute of certain configurations. For example, if there is a box containing five apples, then "fiveness" is a property of the box and the apples taken collectively. In a primitive sense, how would we quantify the apples in the box? By "counting" them with our fingers. We match each apple with one of our fingers. If we run out of apples before we run out of fingers there are less apples then fingers and if we run out of fingers before apples then there are more apples than fingers. If we run out of apples at the same time that we run out of fingers then there are the same number of apples and fingers. This is exactly what occurs mathematically in comparing the cardinality of two sets. Two sets have the same number of elements if there exists a one-to-one onto function (a bijection) that maps one set on to the other. Eventually we give names to these quantities that we intuited (one, two, three, four, etc.). It seems to me that the practitioners of a language would need to have developed well above the level of a small primitive tribe before they would invent words for very large numbers such as million or billion. Even today we abandon the process of inventing new words for very large numbers and use scientific notation with powers of ten giving us the ability to represent arbitrarily large numbers, but when we do so we lose any intuitive understanding of the quantity that such numbers represent. For example, no one has any conception of how 1010000 and 1010001 differ in quantity since there is nothing in nature to compare them with. Though these two numbers can be represented, we are forced to revert to primitive notions of cardinality and the most that we can say is that the second number is bigger than the first. (You need more fingers to count the second than the first. A lot more.)
If quantity (and hence numbers) is something in nature that can be perceived by the human mind, is that capability unique to humans? I think not. Consider a wolf (not a domesticated animal to avoid influence by man) with a liter of five pups. She moves a short distance away and her pups naturally follow but one of them straggles. She suddenly recognizes that one is missing so she backtracks, finds the straggler, resumes her move with all five now in tow. Clearly to recognize that one is missing she must in some sense be counting. Even if one attempts to evade the argument by saying she's not really counting but perhaps had stored in her brain a collection of the scents of her five pups and simple matches those stored scents with the four pups that are present and thus discovers that one is missing, but she doesn't really have any conception of the idea of "fiveness" so she's not really counting. But if she can recognize that one of the five stored scents is not matched up she must in some manner be engaging in some type of cardinality comparison, primitive as it may be. I conclude therefore, that numbers or quantity are inherent and inseparable from nature and are not something created by the human mind but something that is perceived by mind.
helios: [of numbers]
| What is their ontological status? |
I believe that is a question that cannot be answered within the framework of naturalism and the metaphysical assumptions of modern science. For me, as a Christian, the answer to the question as to why the natural world and the Universe are the way they are, as to why the human mind can perceive such order and structure in the Universe as revealed by science, can only be found in the Creator and beyond that it is impossible to push.
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argh, too much to read (so I'm just blithely jumping in)
Numbers are truth - knowledge of things as they are.
Sure, I can hold up five fingers, but the knowledge that I am holding five fingers is usefully important.
Don't forget that truth exists outside <any sentient being's> ability to comprehend it. It simply is.
That is, there is no space where numbers don't work. They do transcend everything.
You don't go far enough. But, nicely put anyway.
As for radix -- we humans tend to like base ten, but that's just convenient for us. The ancient Mayan's thought twenty was pretty neat. (That was the radix of their number system.) So the idea of Klingon's base three system isn't too hard to digest, actually. (They'd just have to write a lot of digits for relatively small numbers... Fortunately Klingons actually are fiction...)
Also
http://xkcd.com/263/
Numbers are truth - knowledge of things as they are.
Sure, I can hold up five fingers, but the knowledge that I am holding five fingers is usefully important.
Don't forget that truth exists outside <any sentient being's> ability to comprehend it. It simply is.
That is, there is no space where numbers don't work. They do transcend everything.
| I conclude therefore, that numbers or quantity are inherent and inseparable from nature and are not something created by the human mind but something that is perceived by mind. |
You don't go far enough. But, nicely put anyway.
As for radix -- we humans tend to like base ten, but that's just convenient for us. The ancient Mayan's thought twenty was pretty neat. (That was the radix of their number system.) So the idea of Klingon's base three system isn't too hard to digest, actually. (They'd just have to write a lot of digits for relatively small numbers... Fortunately Klingons actually are fiction...)
Also
http://xkcd.com/263/
Well, I said they were a mathematical construction, not a human one. My opinion on the matter is that they exist independently of any entity.
nit: Numbers are, in fact, a human construction. It is mathematics that exist independent of any entity. (Hence the point about radix.)
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| Duoas wrote: |
|---|
| nit: Numbers are, in fact, a human construction. It is mathematics that exist independent of any entity. (Hence the point about radix.) |
I also don’t understand the distinction, how could mathematics exist without the concept of numbers? Isn’t that like saying language can exist without grammar?