Are Numbers Real?

[polymath257]

The answer to the question depends on what you mean by real.

Without opening a whole new can of worms, let's just put the question like this: Is math something we create or something we discover? If it is something that we discover, then that implies it has "an existence" of some sort or another, independent of our perceptions.

I suppose another way of framing the question could be: Does math make truth statements? That is, does math make objectively verifiable claims? It could be argued that it does

This is a topic I have thought about A LOT. I think we both invent and discover math. Let me see if I can explain.
Suppose I ask if the game of chess was invented or discovered. I think we all can agree it was invented. But, suppose I ask if, from a particular position in chess, there is mate in 4 moves. That is a question about some truth concerning those invented rules. And we can discover such truths even though the game of chess was itself invented.

Math is primarily an investigation of abstract formal systems. In such systems, we have axioms (basic assumptions). We choose those axioms, thereby inventing a topic in mathematics. Once those axioms have been chosen, however, we discover the consequences of those axioms.

So, that right triangles obey the Pythagorean equality is a discovery from the invented system of Euclidean geometry. If you choose other axioms, say those of non-Euclidean geometry, the Pythagorean equality would fail. There is then no 3-4-5 triangle.
The number pi can be defined in several very different ways, depending on the assumptions being made. In non-Euclidean geometry, though, it is no longer the ratio between the circumference and the diameter of a circle because there is no one such ratio, but many, depending on the size of the circle. The axiom system makes a difference in the truths. More clearly, truth depends on the assumptions made.

The same goes for numbers. We have some basic intuitions concerning numbers and such things as addition and multiplication. Those basic intuitions help us choose our axiom systems, thereby inventing a subject in mathematics. We can then discover whether certain results follow from those axioms. So, for example, in ordinary arithmetic, 13 is a prime number. But, if you use Gaussian integers, it is no longer prime. There is unique factorization into primes for ordinary arithmetic, but not if you look at certain algebraic number fields.

Each system of axioms has statements that it can prove, statements it can falsify (hopefully no overlap as that gives an inconsistency), and statements that it cannot decide. The latter class of statements can either be asserted or denied in conjunction with the other axioms and still have a system that is just as consistent. We get to choose in this case, based on our intuitions and our sense of aesthetics.

So, the answer to your question is that math is invented in that we choose our axioms. But after we do so, the consequences are discovered. Different systems will give different 'truths'.

Which system is best for describing the 'real world' is yet to be determined. That is a matter of experimentation and observation.

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The definition of a circle is, via Euclid, "A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its center."

That is already the ideal, we don't draw anything else from a simple definition.

Pi is simply the ratio of the diameter to the circumference; it just happens to be a transcendental number, which means that it is a decimal fraction that never repeats. It is not an approximation; we use approximate values for it depending on the required precision for what we are doing.

That definition only works in Euclidean geometry. In other geometries, there may not be only one such ratio. Nonetheless, we can define pi in several other ways and reach the same number. But in any case, we have to assume properties of the 'real' numbers to even define what the circumference is supposed to mean, let alone what the possible ratios are.

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We create maths. We discover ways in which particular maths has applications when applied to reality.

Remember that maths need not have any bearing on reality at all. To say that you "discover" a new mathematical system would be metaphorical only; it would be like exploring the platonic plane of abstract concepts.

Maths makes statements which are either true or false within their own framework. They are true, essentially, because we say they are true. They are a logical result of applying the rules which we say are true. Verifying they are true is a matter of making sure the rules have been correctly followed.

So (concerning right triangles) it is only true that the square of the hypotenuse is equal to the square of the sum of the other two sides because we say it's true. This is not a fact that we discovered about right triangles? Is this what you're saying?

It's more that it is true because we say certain axioms are true. No, it is not a fact about all right triangles. As an example, there is a triangle on a sphere which has three right angles: pole to equator, 90 degrees along the equator, then back up to the pole. That is a perfectly legitimate triangle in spherical geometry. And the sum of its angles is 270 degrees, not 180. In such a triangle, even defining which is the hypothesis is problematic (it is a equilateral triangle after all). And the Pythagorean equality fails badly.

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The example of a cricle was given earlier, and no such thing as a circle exists in the so-called real world.  Our idea of a circle is an idealization that is largely a byproduct of the way our senses work, specifically with regard to granularity and sub-feature processing in the brain and eye

If our concept of circularity is based on an idealization what is it an idealization of? Is the value of pi just a rough approximation based on empirical observation? Is there any possible universe in which the value of pi is different?

Depends on your definition of the number pi. Different potential definitions can give different results, depending on underlying assumptions.

Pi is a real number, which means it is an 'object' in an abstract formal system. That system is defined by its axioms and rules of inference. One 'nice' definition of pi is 4 times the integral from 0 to 1 of 1/(1+x^2). No circles need be harmed in this definition. but to define the integral requires a fair amount of work if you are starting from the axioms. But to even get the real numbers requires some work if you are starting from the axioms of set theory.

Now, the usual abstract formal system we use has shown itself useful for making models in physics. That is why that model is preferred.

[Fireball]

Heh, I specifically used Euclid to keep it easy. Some folk get a little glazed in the eyes even at plane geometry. I didn't want to bring spherical geometry in, but you've let the cat out of the bag!

[Angrboda]

Without opening a whole new can of worms, let's just put the question like this: Is math something we create or something we discover? If it is something that we discover, then that implies it has "an existence" of some sort or another, independent of our perceptions.

I suppose another way of framing the question could be: Does math make truth statements? That is, does math make objectively verifiable claims? It could be argued that it does

This is a topic I have thought about A LOT. I think we both invent and discover math. Let me see if I can explain.
Suppose I ask if the game of chess was invented or discovered. I think we all can agree it was invented. But, suppose I ask if, from a particular position in chess, there is mate in 4 moves. That is a question about some truth concerning those invented rules. And we can discover such truths even though the game of chess was itself invented.

Math is primarily an investigation of abstract formal systems. In such systems, we have axioms (basic assumptions). We choose those axioms, thereby inventing a topic in mathematics. Once those axioms have been chosen, however, we discover the consequences of those axioms.

So, that right triangles obey the Pythagorean equality is a discovery from the invented system of Euclidean geometry. If you choose other axioms, say those of non-Euclidean geometry, the Pythagorean equality would fail. There is then no 3-4-5 triangle.
The number pi can be defined in several very different ways, depending on the assumptions being made. In non-Euclidean geometry, though, it is no longer the ratio between the circumference and the diameter of a circle because there is no one such ratio, but many, depending on the size of the circle. The axiom system makes a difference in the truths. More clearly, truth depends on the assumptions made.

The same goes for numbers. We have some basic intuitions concerning numbers and such things as addition and multiplication. Those basic intuitions help us choose our axiom systems, thereby inventing a subject in mathematics. We can then discover whether certain results follow from those axioms. So, for example, in ordinary arithmetic, 13 is a prime number. But, if you use Gaussian integers, it is no longer prime. There is unique factorization into primes for ordinary arithmetic, but not if you look at certain algebraic number fields.

Each system of axioms has statements that it can prove, statements it can falsify (hopefully no overlap as that gives an inconsistency), and statements that it cannot decide. The latter class of statements can either be asserted or denied in conjunction with the other axioms and still have a system that is just as consistent. We get to choose in this case, based on our intuitions and our sense of aesthetics.

So, the answer to your question is that math is invented in that we choose our axioms. But after we do so, the consequences are discovered. Different systems will give different 'truths'.

Which system is best for describing the 'real world' is yet to be determined. That is a matter of experimentation and observation.

It seems that from your examples you are suggesting that mathematics is simply the analytical content of the axioms that we choose. That seems fine as far as it goes, but in cases such as that of the Dirac quote below, we seem to somehow pack an extraordinary amount of analytical truth into a small number of axioms, from which such inferences as those that are made seems to correlate well with the real world almost in anticipation of real world truths. There is nothing necessarily contradicting the possibility that we could choose such bountiful axioms so simply and easily, but it seems to bugger the imagination that we have done so purely by chance. I know there is a good deal of fitting the axioms to the macroscopic reality of the world in mathematics today, but if you go back to, say, Euclid, and his basic postulates of geometry, so much that we can correlate at least to a reasonable approximation flows from that small set of axioms. True, when you take things like the curvature of space-time into account, or possible non-Euclidean spaces, then derivations based upon those axioms will diverge somewhat, yet for our everyday macroscopic world, they seem more than coincidentally descriptive, and our ability to pack so much into such a small number of axioms seems almost magical.

It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.

~ P.A.M. Dirac

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The example of a cricle was given earlier, and no such thing as a circle exists in the so-called real world.  Our idea of a circle is an idealization that is largely a byproduct of the way our senses work, specifically with regard to granularity and sub-feature processing in the brain and eye

If our concept of circularity is based on an idealization what is it an idealization of?

By idealization I mean that it is the maximization of a specific idea. It is not the maximization of things existing in the world but the maximization of other purely constructed mental things like lines, points, curvature, and so forth. In the real world, these ideas are extracted out from data that doesn't actually contain such things, so they aren't an idealization "of" anything, they are just pure mental constructs.

Is the value of pi just a rough approximation based on empirical observation?

No, the value of Pi characterizes a relationship which can be conceived in the mind. We might derive an approximation by drawing an analogy between things which have a similarity to the mental construct, but the value of Pi as it is currently understood is purely a function of that abstraction. You could connect the dots lying along a circle as making up a fractal pattern which, like a coastline, is far longer than the metric of measuring the distance between any two points lying on the circle and the value of the ratio between the radius and the circumference of that figure would be many times greater than Pi. We 'construct' the circle as a circle in our minds by constructing a figure that consists of the separate points lying along the circle as connected by straight lines as a byproduct of our perceptual processes. We "fill in" the missing information to make a correspondence between a set of points and the idea in our mind. However, this operation is highly dependent upon the stimulus and how our perceptual systems process it. If there are 5,000 points along a circle of 6" diameter, our perception has little difficulty in reconstructing those points as belonging to a circle. If you reduce the number of points lying along a circle to five, then our perception doesn't know whether to construct our idea of the object as a circle or as a pentagon. Between these two extremes there will be a crossover region where the perception is more likely to result in the idea of a circle than a non-circle polygon, and where that region occurs depends upon the granularity of our perceptual systems. If the 5,000 points were spread around the periphery of a figure that was as far across as the physical universe is, we would not be as ready to conclude that they necessarily were a part of a circle rather than some other regular or irregular shape.

Is there any possible universe in which the value of pi is different?

Given that Pi is a relationship of a purely mental construct, it doesn't make sense to ask if there is a possible universe in which the value of Pi is different because the value of Pi doesn't exist in this universe. It exists as an analytical truth that follows from the concept of a circle which our mind constructs. If you're asking if we could have a different psychology, I'm sure we could have, but it's not a very meaningful question. I think what you are likely trying to ask is, if we assume that planar geometry accurately describes our universe, are there possible universes in which analogs to the idea in our mind, such as the drawing of a circle on a piece of paper, have different relationships, and the answer to that is yes along several possible dimensions. As pointed out already, we could have a different psychology such that the idea of a circle never resulted from an experience of seemingly contiguous points. I suspect there are evolutionary reasons why we have the psychology we do, but we certainly aren't required to have it. Second, there are geometries in which the ratio of an idealized circle's circumference to its radius is different than Pi. There is nothing that says that a possible universe cannot have a space-time which corresponds to one of these geometries, so the answer along that dimension is yes as well. Then, as noted above, our perceiving a set of points in the real world as lying along a circle depends upon our constructing a figure by drawing lines between the adjacent points along the figure's circumference. That is a simplifying assumption which is hardwired into our perception. It is not a necessary assumption, so as in the case of the pentagon and the conception of the points being perceived as lying along a fractal coastline, neither case would correspond to what we ordinarily give the value of Pi to be, so not only is it possible for there to be other values of Pi in other possible universes, it's possible to have different values of Pi within this universe itself, as our value of Pi depends upon a constructive assumption, and not upon some objective feature of the world.

[robvalue]

Naturally, the axioms of the most used mathematical systems weren’t chosen by accident. They’re essentially the result of reverse engineering from the evolution of applied maths. Wondering at how amazing it is that the maths we mainly use just so happens to model things so well is a bit like when theists make improbability arguments against evolution. The maths has been self-selected because it is useful. As we've become more able to think abstractly, we've formalised the primitive ideas and finally reached the starting point, ironically enough! We didn't start by randomly selecting axioms.

For those not familiar, axioms end up being some statements that are required to be true in order to make the maths work, but which can’t be proven to be true. Ending up with as few axioms as possible is desirable. From what I remember, it’s often the case that you can pick (for example) any 2 out of a particular 3 statements as axioms, and together they will prove the third; but if you pick only 1, it’s not sufficient.

In a general setting, as soon as you pick a group of axioms (along with defining the elements allowed into your system and the operations that can be performed), you’ve instantly defined everything that can be proved by those axioms. It’s just a matter of figuring it all out from there, and then seeing if the results have any useful real-world applications.

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PS: Referring to what Jorm was saying, I don’t have a formal definition for what is "objectively real", because I’ve found it to always be a circular endeavour. I only deal with subjective and relative levels of real-ness. Obviously when talking informally, I use "real" to mean "part of a hypothetical objective reality". So maths isn’t the same kind of real as the supposed physical world; that’s as far as I would go.

[vulcanlogician]

It's more that it is true because we say certain axioms are true. No, it is not a fact about all right triangles. As an example, there is a triangle on a sphere which has three right angles: pole to equator, 90 degrees along the equator, then back up to the pole. That is a perfectly legitimate triangle in spherical geometry. And the sum of its angles is 270 degrees, not 180. In such a triangle, even defining which is the hypothesis is problematic (it is a equilateral triangle after all). And the Pythagorean equality fails badly.

Thank you for your post. This sheds quite a bit of light on things. It is like Rob was saying before: math is only true because its underlying axioms are assumed to be true. But I don't think this presents a problem from mathematical Platonism or moral objectivism (not that I endorse mathematical Platonism or anything, just pointing this out).

What is an axiom? Let's define it as a basic assumption for the purposes of this discussion. I'm sure there is a more precise definition to be had among mathematicians, but that's pretty much what it means in philosophy and (I'm guessing) that's pretty much what it means in math. But a basic assumption can be correct, can it not?

I'm going a bit out of my wheelhouse here (and correct me if I'm wrong) but Euclidean geometry assumes that all its calculations transpire in flat 2D or 3D space. Those basic assumptions were so successful because, by and large, a great deal of the physical world conforms to those assumptions. My point (in the other thread) was that morality is an objective enterprise, just like math. In math, we didn't "choose" the axioms upon which any given system is based out of thin air. There was good reason for our assumptions... at the time when those systems were formulated, their assumptions were considered to be universal (of course they they aren't technically.... but pretty close).

In regards to Jorm's post above, the success of mathematics seems to indicate that (at least some of) our basic assumptions (axioms) are correct. Otherwise, we got quite lucky in selecting them.

What I would disagree with concerning Rob is this:

It’s more like every single person draws up their own moral (mathematical) system, and so what is true in one system is not true in another. It just so happens that certain mathematical systems are so incredibly useful that it’s highly practical to all use the same one in most applied tasks.

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Maths applied through science can give us data and predict outcomes, but it can’t tell us which outcomes are preferable without also including exact criteria for what "preferable" means. It can’t do the ethics for you.

This seems to say

A) People are given a wide berth in selecting axioms, as if they can just pick whichever ones they want. I mean... they can... and the math will still work (I get that). But given the success of certain systems of mathematics in describing the physical world, this seems to suggest that we selected the "right" ones concerning those systems--ie. some of our basic assumptions were correct.

B) That math is "useful" in physics in the same way that the myth of Santa Clause is useful in our ethical discourses with our children. That is, it's a useful story on a practical level, but it is otherwise made up. Rob seems to say, as mathematical fictionalists propose, math does not make truth statements. What is your take on Rob's post?

I would posit that (in ethics as well as math) a truth statement based upon certain axioms is a truth statement nonetheless... provided the basic assumptions (axioms) are correct. But I'm learning quite a bit here, so I'm going to stand back and listen a bit more before advancing any new claims.

[I_am_not_mafia]

But now I feel like we are getting away from the significance of the question in the first place. To follow your and Rob's skepticism any further, we may as well say "there are no such things as facts, because for there to be facts in the first place relies on a basic set of assumptions."

The hive mind aliens may not be able/willing to understand math the way we do, but this has no bearing on the fact that math is a real, objective thing.

But Maths isn't a real or objective thing. It's a made up system.

It's like saying musical notation is real because octaves can be seen having double the frequency of a real world string. But we made the decision that the note A is 440hz and to partition the scale up the way we have even though there are real physical reasons for doing so (like 4th and 5th notes being more in phase than the 7th note in a scale).

For example, what is 0 divided by 0 ?

If I find that I have a problem that cannot be adequately described by Maths, I am free to create a new form of Maths to allow me to reason about it. It's just a tool I use. I'm also not very good at Maths so the system I come up with may not be that effective.


People argue the same argument for logic and claim that it also objectively exists. But this assumes that both the values True and False exist in reality.

They don't.

The only difference with logic is that it is constrained to two values whereas Maths uses an infinite number of numbers.

Unless you use fuzzy logic that is which was created to have the best of both systems.

Or take probability. Is that based on a real underlying chance of something happening, or is it based on our lack of knowledge and understanding of what we predict?

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Mat, I wouldn't imagine that to be the case with aliens.

The periodic table has beautifully simple structures ... eg: Hydrogen = 1 proton, Helium - 2 protons, Lithium = 3 protons., etc

I believe any advanced race will discover all these correlations via maths.

They may recognise that everything that we see as distinct and singular is in fact made up of constituent parts. Much like protons and neutrons in fact being made up of subatomic particles.

[polymath257]

This is a topic I have thought about A LOT. I think we both invent and discover math. Let me see if I can explain.
Suppose I ask if the game of chess was invented or discovered. I think we all can agree it was invented. But, suppose I ask if, from a particular position in chess, there is mate in 4 moves. That is a question about some truth concerning those invented rules. And we can discover such truths even though the game of chess was itself invented.

Math is primarily an investigation of abstract formal systems. In such systems, we have axioms (basic assumptions). We choose those axioms, thereby inventing a topic in mathematics. Once those axioms have been chosen, however, we discover the consequences of those axioms.

So, that right triangles obey the Pythagorean equality is a discovery from the invented system of Euclidean geometry. If you choose other axioms, say those of non-Euclidean geometry, the Pythagorean equality would fail. There is then no 3-4-5 triangle.
The number pi can be defined in several very different ways, depending on the assumptions being made. In non-Euclidean geometry, though, it is no longer the ratio between the circumference and the diameter of a circle because there is no one such ratio, but many, depending on the size of the circle. The axiom system makes a difference in the truths. More clearly, truth depends on the assumptions made.

The same goes for numbers. We have some basic intuitions concerning numbers and such things as addition and multiplication. Those basic intuitions help us choose our axiom systems, thereby inventing a subject in mathematics. We can then discover whether certain results follow from those axioms. So, for example, in ordinary arithmetic, 13 is a prime number. But, if you use Gaussian integers, it is no longer prime. There is unique factorization into primes for ordinary arithmetic, but not if you look at certain algebraic number fields.

Each system of axioms has statements that it can prove, statements it can falsify (hopefully no overlap as that gives an inconsistency), and statements that it cannot decide. The latter class of statements can either be asserted or denied in conjunction with the other axioms and still have a system that is just as consistent. We get to choose in this case, based on our intuitions and our sense of aesthetics.

So, the answer to your question is that math is invented in that we choose our axioms. But after we do so, the consequences are discovered. Different systems will give different 'truths'.

Which system is best for describing the 'real world' is yet to be determined. That is a matter of experimentation and observation.

It seems that from your examples you are suggesting that mathematics is simply the analytical content of the axioms that we choose.  That seems fine as far as it goes, but in cases such as that of the Dirac quote below, we seem to somehow pack an extraordinary amount of analytical truth into a small number of axioms, from which such inferences as those that are made seems to correlate well with the real world almost in anticipation of real world truths.  There is nothing necessarily contradicting the possibility that we could choose such bountiful axioms so simply and easily, but it seems to bugger the imagination that we have done so purely by chance.  I know there is a good deal of fitting the axioms to the macroscopic reality of the world in mathematics today, but if you go back to, say, Euclid, and his basic postulates of geometry, so much that we can correlate at least to a reasonable approximation flows from that small set of axioms.  True, when you take things like the curvature of space-time into account, or possible non-Euclidean spaces, then derivations based upon those axioms will diverge somewhat, yet for our everyday macroscopic world, they seem more than coincidentally descriptive, and our ability to pack so much into such a small number of axioms seems almost magical.

It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.

~ P.A.M. Dirac

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Well, Euclid was the end result of a LOT of mathematical investigation. He took what had been 'applied math' used by the Egyptians and 'Babylonians' and combined it with some of the more theoretical material done by the Pythagoreans and Theatetus to figure out which axioms would do what he wanted to do. I would also point out that Euclid missed some assumptions that he nonetheless used implicitly concerning notions of betweenness.


So, no, the axioms are chosen, in almost every case, to describe some intuition that we have. At his point in time, even if the current axioms were found to be inconsistent (which Godel's results allow), there are certain mathematical statements what would survive into any new axiom system. For example, some version of the fundamental theorem of calculus will survive: it has simply been too useful for making models of physics. Many aspects of number theory would survive. But it is quite possible that large tracts of topology would go away.


So you are right, the axioms are NOT chosen by chance. They are chose specifically to abide by our intuitions and to be able to derive certain central results. Currently, all of mathematics is based on about a dozen axioms for set theory (the axioms of Zormelo and Fraenkl). Almost all the rest of mathematics can be expressed within those axioms (the exceptions are certain aspects of proper classes and extensions of the ZF axioms). These axioms were chosen and largely agreed to at the beginning of the last century.


And it *is* interesting that such a small number of axioms can be as expressive as they are. Again, this is partly why those axioms were chosen. The ability to model physical situations is another goal, although more often expressed as the ability to produce certain areas of math like differential geometry.


When applied to the 'real world', math becomes a language with unique expressive power. Because we get to define axioms however we want, we can adjust to any situation that we observe. This vastly increases the possibility of finding a model that works. Furthermore, it is often the case that mathematicians, in their curiosity, have already explored the systems that can later become useful for the physicists. This has lead to the claim that there is an 'unreasonable effectiveness to math'.
But this ignores the vast amounts of math that have *nothing* to do with the real world. Those that have little or no bearing on any question likely to arise in the real world. These areas of math are studied by mathematicians for their beauty, not for their applications. And those areas don't seem to give surprising new correlations with new physics.

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It's more that it is true because we say certain axioms are true. No, it is not a fact about all right triangles. As an example, there is a triangle on a sphere which has three right angles: pole to equator, 90 degrees along the equator, then back up to the pole. That is a perfectly legitimate triangle in spherical geometry. And the sum of its angles is 270 degrees, not 180. In such a triangle, even defining which is the hypothesis is problematic (it is a equilateral triangle after all). And the Pythagorean equality fails badly.

Thank you for your post. This sheds quite a bit of light on things. It is like Rob was saying before: math is only true because its underlying axioms are assumed to be true. But I don't think this presents a problem from mathematical Platonism or moral objectivism (not that I endorse mathematical Platonism or anything, just pointing this out).

What is an axiom? Let's define it as a basic assumption for the purposes of this discussion. I'm sure there is a more precise definition to be had among mathematicians, but that's pretty much what it means in philosophy and (I'm guessing) that's pretty much what it means in math. But a basic assumption can be correct, can it not?

I'm going a bit out of my wheelhouse here (and correct me if I'm wrong) but Euclidean geometry assumes that all its calculations transpire in flat 2D or 3D space. Those basic assumptions were so successful because, by and large, a great deal of the physical world conforms to those assumptions. My point (in the other thread) was that morality is an objective enterprise, just like math. In math, we didn't "choose" the axioms upon which any given system is based out of thin air. There was good reason for our assumptions... at the time when those systems were formulated, their assumptions were considered to be universal (of course they they aren't technically.... but pretty close).

The more modern viewpoint is that 'flat 2d and 3d space' are to be defined as those satisfying those axioms. They are very good approximations, yes. But once you start having to take the curvature of the Earth into account, it is better to use spherical geometry.

In regards to Jorm's post above, the success of mathematics seems to indicate that (at least some of) our basic assumptions (axioms) are correct. Otherwise, we got quite lucky in selecting them.

What I would disagree with concerning Rob is this:

It’s more like every single person draws up their own moral (mathematical) system, and so what is true in one system is not true in another. It just so happens that certain mathematical systems are so incredibly useful that it’s highly practical to all use the same one in most applied tasks.

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Maths applied through science can give us data and predict outcomes, but it can’t tell us which outcomes are preferable without also including exact criteria for what "preferable" means. It can’t do the ethics for you.

This seems to say

A) People are given a wide berth in selecting axioms, as if they can just pick whichever ones they want. I mean... they can... and the math will still work (I get that). But given the success of certain systems of mathematics in describing the physical world, this seems to suggest that we selected the "right" ones concerning those systems--ie. some of our basic assumptions were correct.  

B) That math is "useful" in physics in the same way that the myth of Santa Clause is useful in our ethical discourses with our children. That is, it's a useful story on a practical level, but it is otherwise made up. Rob seems to say, as mathematical fictionalists propose, math does not make truth statements. What is your take on Rob's post?

I would posit that (in ethics as well as math) a truth statement based upon certain axioms is a truth statement nonetheless... provided the basic assumptions (axioms) are correct. But I'm learning quite a bit here, so I'm going to stand back and listen a bit more before advancing any new claims.

I wouldn't say our assumptions were *correct* so much as they were *useful*. But they were originally selected *because* they were useful, so that isn't too surprising.

As for B). I see math as more of a language when it is applied than anything else. Because of how math is, it allows a great deal of flexibility in describing things. So, the axioms are generally built in such a way that we have a great deal of expressiveness in the system. They aren't so much assumptions as means of fitting things together to express what we observe.

But, and this is important (I think), there is a LOT of math that has nothing to do at all with the real world and modeling it. Nobody expects it to be relevant to and scientific investigation. It is studied only because mathematicians (such as we are) find the structures to be beautiful.

[Sal]

I think numbers are representations of objects, i.e. nominally existing. In OPs vid, there's a questionmark for pi when discussing nominalism, seems to me that's easily solved by simply saying it's the ratio between the circumference of a circle to its diameter. This is not hard.

Something similar can probably be said about imaginary numbers, just more difficult than pi.

[robvalue]

I’d say they are representations of quantities of objects. Quantity itself is an abstract concept we use to help us make sense of reality. This is when considering applied standard maths. In pure maths, or other maths systems, the numbers need not represent anything.

Then again, pi becomes rather nebulous as a "quantity". Like you say, numbers are sometimes best viewed as ratios between real values such as distance.

(Edit: representations of dimensions is another way to think of numbers, although you may have a strange situation where certain amounts aren’t physically possible in the real world due to stable wavelengths / sizes of particles and so on.)

[Dr H]

What say you? Are numbers real? If so, in what way are they real?

First answer:

Are you real?
Not facetious:  it raises the issue of defining "real".


Second answer:

Some numbers are indeed real.
Others are imaginary.
Others are complex.

<ducks>