So, metaphysics then.

[JimC]

He was dodging the point. Read his earlier stuff. He is clear that he thinks numbers are somewhat equivalent to physical objects.

Did he actually say that "numbers are somewhat equivalent to physical objects"? I don't recall reading anything of the sort. Best leave it to Jim to clarify now.

Numbers can be abstractions, certainly, but they are also empirical facts about sets of objects. If they were only abstractions, then there would be no difference between a set of 9 pebbles and a set of 8 pebbles, on a riverbed no sapient being has ever observed, and therefore abstracted those numbers.

If a collection of physical objects is real, whether it is perceived by an observer or not, then so is every detail about them; the frequencies of light they reflect, their volume, their mass, the number of protons they contain, and finally, the number of discrete entities they represent. (Hopefully, we can agree that at least some physical entities (like pebbles) have clear and obvious boundaries, making a count of their number non-problematic).

Are those physical properties of a set of objects real, even if they have not been observed by a sentient being? I argue that they are real, and in fact they (along with many other details about our putative pebbles) make up their physical reality, measured or not. If so, then the count (a number) of the set of pebbles has the same degree of reality.

The true abstraction emerges when we manipulate numbers which are not counts of actual sets of objects in the real world.

[JimC]

The 'problem' is that it's not clear what a unity is in regards to counting. Is a cloud 1 object, or is it multiple objects?

That is no more of a problem than defining a crowd. Earth is crowded with bazillions of bacteria and trillions of insects. There was a crowd of 175,000 people at the recent Glastonbury festival. A bus might be crowded with 20 people. Three's a crowd. A metaphysical problem might be crowding my mind. A cloud might be a barely perceptible whisp of water droplets or a humongous bank of an advancing thunderstorm. Units are rarely precisely defined, and even they may change over time. The metre used to be defined by some length of a piece of very inert metal resting in a Paris museum. Now it is defined by x vibrations per second of some crystal.

Actually, its definition now is "the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second."

I think you were confusing its definition with that of the second, which is defined as "the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom."

Some interesting issues here. Now, given that we are talking about a quantum transition, this will involve a discrete amount of energy, which should be fixed, not variable. Measured in appropriate units, will the number involved be rational or irrational? The amount of energy will determine the wavelength of the photon released, and hence its frequency and period. Again, will these figures be rational (possibly even integers, if we are talking a frequency), or irrational?

I'm actually not sure...

Surely, whether the number is rational or not depends on the units.

Wavelength = hc/E, where h is Planck's constant, c is the speed of light and E is the energy of the photon. Pick appropriate units for these and you can make any wavelength rational, or even an integer. But that doesn't really make it special. Just as the metre and the second aren't special - just arbitrary choices.

I agree that units are arbitrary, but I'm not sure that the choice of units can alter whether a unit is rational or irrational (integer vs fractional, sure...)

For example, take a 3 metre by 4 metre by 5 metre right angled triangle. All its sides are rational numbers, and they would be in any other unit of length you choose.

But it we take a right angled triangle with the 2 short sides 1 metre each, then the hypotenuse would be root 2 metres. If you converted to any other unit of length, the 2 short sides would still be rational (albeit non-integer) numbers, and the hypotenuse would be still irrational.

I think this example from length generalises to other measurements, in principle.

But I was thinking of other issues. For example, we have measured certain physical quantities such as the mass of an electron to a certain number of decimal places, the current limit, I presume, being controlled by the current precision of the apparatus involved. Is that number in principle an irrational number like Pi, or could it be a rational number that we have not yet exactly enumerated? Are there real, physical qualities in the world which are truly irrational numbers, or are they all potentially rational, or are there some of each?

[JimC]

Perhaps I'm getting too impressed with the difference between rational & irrational numbers. Both of them represent a point on the real number line, maybe that's all that matters...

Yet again, perhaps no physical quantity in the actual universe is represented by a unique point on the real number line, which implies an infinite number of decimal places, and therefore an infinite degree of precision, which should be impossible due to the uncertainty principle...

My head hurts...

[MiM]

Let us for a second contemplate an universe, where no discrete entities exists. We may think of a huge ball of plasma. Let us further contemplate an intelligence in this universe, an intelligence that would not be easily defined as "one" or "many" individuals, more like a thought svirling around with the plasma. Would this intelligence be likely to ever think in form of numbers? would it have the concept of discrete mathematics? I think not. Would it have superb concepts for mathematically describing fluid motion, I bet!

This is the way of thinking that leads me to hold numbers as more than just abstract concepts. They are extremely strongly linked to the way our physical universe is shaped.

[Scott1328]

For example, take a 3 metre by 4 metre by 5 metre right angled triangle. All its sides are rational numbers, and they would be in any other unit of length you choose.

But it we take a right angled triangle with the 2 short sides 1 metre each, then the hypotenuse would be root 2 metres. If you converted to any other unit of length, the 2 short sides would still be rational (albeit non-integer) numbers, and the hypotenuse would be still irrational.

I think this example from length generalises to other measurements, in principle.

But I was thinking of other issues. For example, we have measured certain physical quantities such as the mass of an electron to a certain number of decimal places, the current limit, I presume, being controlled by the current precision of the apparatus involved. Is that number in principle an irrational number like Pi, or could it be a rational number that we have not yet exactly enumerated? Are there real, physical qualities in the world which are truly irrational numbers, or are they all potentially rational, or are there some of each?

It is not known if the universe is continuous or discrete. What is known is that is impossible to make any measurement with a precision finer than the Planck length. This then means that in this universe it is actually impossible to demonstrate any lengths other than rational lengths. And since this is the case, why does it make any sense to say that irrational numbers exist other than as abstractions?

[Xamonas Chegwé]

Jim, you only need to compare radians and degrees. 360º is an integer, 2π radians is not.

Your triangle could be 3π knips by 4π knips by 5π knips in another scheme where 1 knip is 1 metre/π. All sides irrational.

What I think you meant was that the ratios between the sides were always rational. This is true. Sine, Cos & Tan of a 3,4,5 triangle are always rational.

[VazScep]

In the really olden days, when folk didn't talk of "irrational numbers", folk talked instead of "incommensurable magnitudes." The difference is that "irrational" is a property of a single number, while "incommensurable" is a realtion between two line segments.

So instead of saying that the square root of 2 is irrational, you would say that the diagonal of a square is incommensurable with the side. And instead of saying that π is irrational, you would say that the diameter of a circle is incommensurable with the circumference.

You can translate freely between the two. The number 1 is rational because it's the line segment you started with, and your benchmark for commensurability. Then when you construct the square whose sides are commensurate with that initial line segment, you find that the diagonal is incommensurate with 1, or, in modern folk-speak, irrational.

That side-steps this stuff about a triangle whose sides are 3π, 4π and 5π, which would be a triangle constructed by first squaring the circle, and then using the resulting edges of the square to construct a triangle, a problem manifestly more difficult than constructing the 3,4,5 triangle.

[JimC]

Scott1328 wrote:

It is not known if the universe is continuous or discrete. What is known is that is impossible to make any measurement with a precision finer than the Planck length. This then means that in this universe it is actually impossible to demonstrate any lengths other than rational lengths. And since this is the case, why does it make any sense to say that irrational numbers exist other than as abstractions?

Good point, Scott. I wonder whether the limitation of measurement below the Planck length actually reflects some form of real discrete nature of the universe, or whether it is actually continuous "to the infinitesimal and beyond", and it's just the continuous "bubbling" of the quantum foam which prevents a definite measurement?

Jim, you only need to compare radians and degrees. 360º is an integer, 2π radians is not.

Your triangle could be 3π knips by 4π knips by 5π knips in another scheme where 1 knip is 1 metre/π. All sides irrational.

What I think you meant was that the ratios between the sides were always rational. This is true. Sine, Cos & Tan of a 3,4,5 triangle are always rational.

Good point about radians vs degrees. But does this rational = irrational number thing happen in any system without π? (mind you, it does crop up all over the place)

In the really olden days, when folk didn't talk of "irrational numbers", folk talked instead of "incommensurable magnitudes." The difference is that "irrational" is a property of a single number, while "incommensurable" is a realtion between two line segments.

So instead of saying that the square root of 2 is irrational, you would say that the diagonal of a square is incommensurable with the side. And instead of saying that π is irrational, you would say that the diameter of a circle is incommensurable with the circumference.

You can translate freely between the two. The number 1 is rational because it's the line segment you started with, and your benchmark for commensurability. Then when you construct the square whose sides are commensurate with that initial line segment, you find that the diagonal is incommensurate with 1, or, in modern folk-speak, irrational.

That side-steps this stuff about a triangle whose sides are 3π, 4π and 5π, which would be a triangle constructed by first squaring the circle, and then using the resulting edges of the square to construct a triangle, a problem manifestly more difficult than constructing the 3,4,5 triangle.

Also good points, although I think XC was envisaging the triangle actually existing, rather than having to be constructed. But perhaps the whole issue revolves around construction, starting with integers, which are the counting numbers that I aver have a physical reality as part of the feature of sets of real objects...

[VazScep]

Also good points, although I think XC was envisaging the triangle actually existing, rather than having to be constructed.

A classic mistake, if you'll forgive the pun. In my not so humble opinion, a triangle only exists in so much as it has a construction.

But perhaps the whole issue revolves around construction, starting with integers, which are the counting numbers that I aver have a physical reality as part of the feature of sets of real objects...

This, I'd say, only pushes the issue onto what a set is, a question which takes pride of place in the foundations of mathematics.

[JimC]

Also good points, although I think XC was envisaging the triangle actually existing, rather than having to be constructed.

A classic mistake, if you'll forgive the pun. In my not so humble opinion, a triangle only exists in so much as it has a construction.

It's the word "exists" that much of this thread hangs on. Certainly there is a set of geometrical shapes that can be constructed in the classic sense, and a set that we can envisage, but cannot be constructed. Perhaps it is tempting to give the first set a special position in terms of being potentially real in the physical universe, I'm not sure...

But in an abstract mathematical "space", any object, set or relation with internal consistency can be "created", analysed and compared to other mathematical entities. At that level, we certainly aren't in Kansas anymore, Toto...

[VazScep]

It's the word "exists" that much of this thread hangs on. Certainly there is a set of geometrical shapes that can be constructed in the classic sense, and a set that we can envisage, but cannot be constructed. Perhaps it is tempting to give the first set a special position in terms of being potentially real in the physical universe, I'm not sure...

Basic logic makes the distinction for us, because potentiality can be formalised by saying "if it were the case that...", in other words, by adding a hypothesis, or making a hypothetical judgement.

The stuff we construct is the stuff that exists. The stuff which classical mathematicians say exists but cannot be constructed is just the stuff that could be constructed if we had access to a suitable hypothetical device.

Here's an example: Archimedes, the guy credited for discovering π, proved that the circle could be squared on the hypothesis that he had access to a means to construct spirals.

But in an abstract mathematical "space", any object, set or relation with internal consistency can be "created", analysed and compared to other mathematical entities. At that level, we certainly aren't in Kansas anymore, Toto...

Even in arbitrary abstract mathematical spaces, analysis and comparison can proceed by constructive means of analysis and comparison.

[mistermack]

While I'm not really up on the subject, I'm not sure that I can go along with the talk of objects.

What is an object? I'm definitely not an object. I'm an event. I'm no more an object than a fire is.
The same goes for practically everything. They are constantly changing over time. And interacting with the rest of the universe.

Everything has got particles whizzing through it all the time.
You can only have an object, if you freeze time. In real time, everything is changing, at various speeds.

What that's got to do with anything, I don't really know.

[piscator]

There are differences between things.
No differences, no math. Therefore, difference-an-sich is a first principle of math, and of any metaphysik.

[DaveDodo007]

Go home seal you're drunk.

[piscator]

Shit. My car was right on the end of this key...