What's your opinion of a Math without Reals?

[FactChecker]

The real world is discrete and finite.

Nonsense. Many human preformed processes may be discrete and finite, but not the real world.
Suppose you measured distances on a line with a given unit measure and could measure all rational distances. You are saying that the irrational distances do not exist in the real world. Pick another randomly generated length as a new unit length. The odds are 0 that it was a rational length in the original units. But now you would say that those lengths exist in the real world and the first set no longer do. That is nonsense at so many levels.

This whole discussion goes back to Pythagoras, who was horrified by the proof that the hypotenuse of a right triangle with two unit sides would be irrational. I hope we have progressed since then.

 

[fresh_42]

Nonsense. Many human preformed processes may be discrete and finite, but not the real world.

You can count molecules and atoms and estimate the finite mass of the universe, can't you? So it's finite and also discrete. A length is nothing more than molecules on a line. The concept of a continuum between two points is already a mental concept that bares a real correspondence. To disqualify it as nonsense doesn't support your argument. Au contraire, it weakens it. But you're right as in so far it doesn't make any sense to debate with you on such a sophisticated level.

 

[FactChecker]

You can count molecules and atoms and estimate the finite mass of the universe, can't you? So it's finite and also discrete. A length is nothing more than molecules on a line. The concept of a continuum between two points is already a mental concept that bares a real correspondence. To disqualify it as nonsense doesn't support your argument. Au contraire, it weakens it. But you're right as in so far it doesn't make any sense to debate with you on such a sophisticated level.

So are you saying that there is no space between molecules? How would you measure an expected space between molecules if you deny the existence of a finer resolution?

 

[jbriggs444]

Unfortunately, there are actual physical items like the circumference of a circle or the hypotenuse of a right triangle with unit sides.

Are there? How would we know one way or the other? To the best of my knowledge, I have never seen either a right triangle or a circle in real life.

 

[votingmachine]

You can count molecules and atoms and estimate the finite mass of the universe, can't you? So it's finite and also discrete. A length is nothing more than molecules on a line. The concept of a continuum between two points is already a mental concept that bares a real correspondence. To disqualify it as nonsense doesn't support your argument. Au contraire, it weakens it. But you're right as in so far it doesn't make any sense to debate with you on such a sophisticated level.

Ok. So pick another property. Perhaps mass is finite, to a perfect, exact measurement. What about energy then? Or tell me what the momentum is of this exactly known mass? QM may require you to re-think the certainty of your principles.

 

[micromass]

Ok. So pick another property. Perhaps mass is finite, to a perfect, exact measurement. What about energy then? Or tell me what the momentum is of this exactly known mass? QM may require you to re-think the certainty of your principles.

But is energy real? Sure, you can make a framework that involves physical quantities and makes extremely accurate predictions (=QM). But that doesn't make me want to say that energy or momentum are actually real things. To me, they are quantities that exist in the mathematical framework, not reality.

 

[micromass]

So are you saying that there is no space between molecules? How would you measure an expected space between molecules if you deny the existence of a finer resolution?

But you can't measure it to arbitrary precision. Maybe that is because our measurement instruments are not good enough, or maybe there is a limited precision. Saying we can measure it to as fine a resolution as we can is already an assumption.

 

[fresh_42]

Ok. So pick another property. Perhaps mass is finite, to a perfect, exact measurement. What about energy then? Or tell me what the momentum is of this exactly known mass? QM may require you to re-think the certainty of your principles.

Everything you measure, is a difference in the configuration of the finite measurement device, and can thus be described by finite means. Everything else happens in our brains, which are again a configuration of finite size. There is no continuum in existence, but it is a perfect tool to describe all these really many things out there. I only claim, that the concept of infinity is man made to handle physics, which we otherwise couldn't. But this doesn't put it into existence. To refuse the usage of real numbers does in my opinion imply to set up science on a finite number of countable things. I admit this number is really large, but it is far from being infinite. Of course this is an extreme point of view. As the avoidance of infinite concepts are. I simply claim, that a continuum is already a creative act. Similar to the usage of $0$ in ancient times.

 

[SSequence]

I disagree that it is a statement of mathematical truth. It is a re-statement of a definition.
...
(a+b)^2=a^2+b^2+2ab
is simply a definition statement of the distributive and commutative laws for defined math functions.

Well think about it this way. You could use rationals and use all these properties (and then scale back to natural numbers or integers). You could use several fairly restricted definitions of reals and use all these properties (and then scale back to natural numbers or integers).

To me all the addition, multiplication, inverse, closure properties are perhaps somewhat trivial. With more restricted definitions, I guess perhaps conceptually difficult part is how to handle the bounding property in a good way? But honestly I don't know much about it though.

edit: Removed the second part of the post because it wasn't well thought out.

 

[FactChecker]

But you can't measure it to arbitrary precision. Maybe that is because our measurement instruments are not good enough, or maybe there is a limited precision. Saying we can measure it to as fine a resolution as we can is already an assumption.

I didn't say you can measure it to an arbitrary precision. I said that it exists to an arbitrary precision. That's a big difference.
Say two people independently decide on a unit measure. The probability is zero that the rational distances in one system are also rational in the other. But they both exist and one is just as valid as the other.

 

[micromass]

I said that it exists to an arbitrary precision.

Does it? Seems like an assumption you make. A reasonable one perhaps, but an assumption nevertheless.

 

[votingmachine]

Are there? How would we know one way or the other? To the best of my knowledge, I have never seen either a right triangle or a circle in real life.

I've never seen a perfect right triangle with sides 3, 4, ad 5 either. But it fits the Pythagorean theorem exactly.

I think this was partly addressed in the debate video. You are confusing the number with the name for the number.

Take the number "1". In english: "one". In spanish "uno". In Czech: "jedna". Written in latin numbers: "i".

We know that it is a number. But is there such a thing? I don't think there is. Certainly I learned to count. If I have a single book, and I count it. I say I have "one" book. I use the abstract concept for the number, that I accept the name "one" for. Yet without a thing available to count, it does not exist.

I'm not confused by the integers. I'm not confused by the real numbers. I don't confuse the electrons that generate the screen output that allows you to understand me, with my thoughts on what a number is.

EVERYTHING we communicate is by necessity an abstraction. Norman Wildberger has a particular issue with abstractions that are difficult to precisely name, in exact symbolic terms, using the digits (not the numbers). Anything that requires assembling a lot of digits is something that he does not believe in. Googolplex is a name for a number. It is a number that is tedious to write down ... make that impossible. But I still have no problem with the number Googolplex existing. It is the number after the number before it.

I'm a bit mystified by this topic. In answer to the original post, I don't see any reason to go down this particular road. I find number such as "the square-root-of-two" to be no more conceptually difficult than a number like "one". Both are easy for me to write down with paper and pencil, and to understand the "number" that I am talking about.

 

[FactChecker]

Does it? Seems like an assumption you make. A reasonable one perhaps, but an assumption nevertheless.

If we say that a right triangle with unit sides exist, how can the length of the hypotenuse not exist? If you say that unit sides do not exist, then we don't even have the natural numbers. The existence of irrational √2 may have upset Pythagoras, but it should not upset us.

 

[micromass]

If we say that a right triangle with unit sides exist

I don't think it does. And if it does, we have no way to check whether it is one.

 

[FactChecker]

I don't think it does.

Suit yourself.

 

[micromass]

Suit yourself.

Sure, you can believe it exists. But it's another assumption you're making. You might not consider it to be a heavy assumption, but it is one nevertheless.

 

[FactChecker]

Sure, you can believe it exists. But it's another assumption you're making. You might not consider it to be a heavy assumption, but it is one nevertheless.

You seem very confident that the existence of a right triangle with 2 equal sides has never been proven. That surprises me.

 

[micromass]

You seem very confident that the existence of a right triangle with 2 equal sides has never been proven.

Do you know of such a proof? I would be glad if you could tell me.

But yes, as long as I don't see the proof or any good argument, I'm remaining agnostic about its existence.

 

[FactChecker]

Do you know of such a proof? I would be glad if you could tell me.

But yes, as long as I don't see the proof or any good argument, I'm remaining agnostic about its existence.

In fact, if I understand you, you are saying that there can not be a rigorous formal mathematical theory where one exists.

 

[micromass]

In fact, if I understand you, you are saying that there can not be a rigorous formal mathematical theory where one exists.

I'm not talking about mathematics. I'm talking about the real world. I have no problem with triangles in mathematics. It forms a useful abstraction of the real world. But it's not reality, only a good approximation.

 

[FactChecker]

I'm not talking about mathematics. I'm talking about the real world. I have no problem with triangles in mathematics. It forms a useful abstraction of the real world. But it's not reality, only a good approximation.

An exact triangle of that nature is no less likely in the real world as any other specific approximation.

 

[FactChecker]

This discussion is getting too silly for me. Bye.

 

[micromass]

An exact triangle of that nature is no less likely in the real world as any other specific approximation.

What makes you think that way? Why would the mathematical formalism imply any existence statement in the real world?

 

[votingmachine]

One does not need a right triangle with unit sides to engage the real number "the square root of two". I simply ask the question:

Is there a number that when squared is two?

I do not require that you be able to write the number as a fraction of two rational numbers. I only require that you invent a symbol that says when squared, the number is 2.

So a valid answer is:
2^(1/2)

We can name this number anything. Call it "the square-root-of-two". Call it Fred. I don't care. If you want to compute a decimal approximation, it will be an infinitely long sequence of digits that need to be computed. If you require that anything real be calculable in a computer, you are making a very narrow requirement. (One that seemingly requires that reality itself be calculable in a computer ... ).

I'm not having any problem with allowing for the existence of a number that is irrational. I am not having any problem with the concept of a number that can never be written as a decimal number EXACTLY. I don't think that Wildberger has stumbled on a particular weakness of math, but rather a weakness of his own.

2^(1/2) IS A REAL NUMBER. If you only want to USE numbers that can be reduced to fractions ... then you cannot answer my question of what number squared is two. You can only say that no rational number can be squared to generate 2. I fail to see any benefit to not "believing" in the real numbers. Or not "believing" in integer numbers that are of extraordinary magnitude.

 

[jbriggs444]

I think this was partly addressed in the debate video. You are confusing the number with the name for the number.

I made a statement about right triangles and circles. You seem to be confusing that with a statement about numbers.

 

[pwsnafu]

An exact triangle of that nature is no less likely in the real world as any other specific approximation.

If you build a right angle triangle with side length one, the probability that it has exact side length one is zero. And yes, that is indeed "no less likely" than all the others, but only because there is nothing less than prob zero. The statement is useless as an argument.

 

[FactChecker]

If you build a right angle triangle with side length one, the probability that it has exact side length one is zero. And yes, that is indeed "no less likely" than all the others, but only because there is nothing less than prob zero. The statement is useless as an argument.

The contention that nature would show a preference for rational numbers that are based on a man-made scale is hard to support. Rational numbers on one scale are irrational numbers on many other scales.

 

[SSequence]

(1) If you require that anything real be calculable in a computer, you are making a very narrow requirement. (One that seemingly requires that reality itself be calculable in a computer ... ).
...
(2) I am not having any problem with the concept of a number that can never be written as a decimal number EXACTLY.
...
(3) I fail to see any benefit to not "believing" in the real numbers. Or not "believing" in integer numbers that are of extraordinary magnitude.

Well, regarding my personal opinion, here are the answer to these three points:
(1) There shouldn't be any relation between the two.

(2) This is a good point. I don't think either that there should be any problem with that either.

The issue is somewhat more subtle. If we just stick "strictly" to decimal representation for example, here is how it goes (at least how I have thought about it):
The problem is that the pessimist interpretation (closer to one an intuitionist will make) is that "perhaps" (but not necessarily) there are only limited number of reals. This is why to seek greater certainty of reasoning we shouldn't use the assumption of unlimited number of reals when reasoning in an absolute sense about statements (note the word "absolute" is very important here).
The optimistic interpretation (closer to one the classical mathematician will make) is that there ARE unlimited number of reals. What this requires is that the possibility of limited number of reals is ruled out entirely and this is where the sense of the difference lies.

The optimist will say I have more tools at my disposal and I can prove more and more stuff without inconsistency. The pessimist will respond that lack of inconsistency does not mean that your reasoning is correct -- indeed it very well may be, but you must show me convincing reasons to rule out the possibility limited number of reals. So until I have that I will just stick to my more modest form of reasoning where only limited number of reals are assumed -- since in that case the statements I prove are more certain.

(3) None of these are issues, at least for me.

P.S.
I am not very well learned in the specifics of this topic (or math in general to be fair). This is my generic personal understanding regarding this topic.

 

[bobkolker]

That would leave the underlying theories of physics without topological qualities. What would near and almost mean. In addition, the theories wold also be devoid of geometric character. That is bad news when one considers that the General Theory of Relativity is a reduction of gravitation to geometric and metrical properties of the space time continuum, which could not be defined in the absences of the real number system.

 

[Mark44]

(2) I am not having any problem with the concept of a number that can never be written as a decimal number EXACTLY.

2) This is a good point. I don't think either that there should be any problem with that either.

The issue is somewhat more subtle. If we just stick "strictly" to decimal representation for example, here is how it goes (at least how I have thought about it):
The problem is that the pessimist interpretation (closer to one an intuitionist will make) is that "perhaps" (but not necessarily) there are only limited number of reals. This is why to seek greater certainty of reasoning we shouldn't use the assumption of unlimited number of reals when reasoning in an absolute sense about statements (note the word "absolute" is very important here).
The optimistic interpretation (closer to one the classical mathematician will make) is that there ARE unlimited number of reals. What this requires is that the possibility of limited number of reals is ruled out entirely and this is where the sense of the difference lies.

With regard to "limited number of reals" -- how are you defining "limited"? Cantor's diagonal argument shows that there an uncountably infinite number of real numbers in the interval [0, 1]. He assumes that these numbers can arranged in a countably infinite list, and then constructs a number that is not on the list, thereby reaching a contradiction.

Possibly there is some question among philosophers about how many real numbers there are, but I don't believe any mathematicians hold the view that the reals are limited.