What's your opinion of a Math without Reals?

[fresh_42]

While real numbers do exist as an abstraction for humans, they do not exist in any way for computers - nor does infinity.

You don't need to build computers on a binary basis. They can as well be analog and this brings in the continuum again.

 

[dkotschessaa]

You don't need to build computers on a binary basis. They can as well be analog and this brings in the continuum again.

Neat. I wonder how this fits into what I know about computability/decidability and such.

-Dave K

 

[epenguin]

So if you have an issue with the reals, you must have an issue with all infinite sets.

He does, and has spoken at length about it. If you look at the titles of his lectures you might find that amongst them. Also if you look at his site you find links to some debates he has sustained. These will mean more to mathematicians than to me, I don't know of any place where he has expressed concisely (concision is not his strong point) the essence of his ideas altogether. Not in fashion accessible to nonmathematicians like me anyway, but what he says may mean more to you because you'd know what he has in mind. (I'd guess you won't be very convinced.) As it is he seems to spread it all out in his many examples. Many of these are quite accessible, and he treats conventional mathematical areas, sometimes with a neat twist I thought (haven't had time to watch very many).

 

[sophiecentaur]

It seems to me that the only problem with this guy's restriction is that it restricts the number of Real-World situations that it can model. It is irrelevant for most of us most of the time. But that's just the same as inventing a board game which doesn't apply to Science; it's not a reason for taking offence.

 

[votingmachine]

Well as far as my personal opinion goes, I not only think that these are mathematically meaningful sets (as I already mentioned) but also a statement such as:
"(a+b)^2=a^2+b^2+2ab -- for all a and b belonging to N={0,1,2,3,...} (or say Z -- set of integers)"
is simply a "genuine" and meaningful mathematical truth and so are statements such as "All primitive recursive functions are total" etc.
(I only used the word "genuine" to emphasize the cases where the statement is plainly easy to assess -- or the line of reasoning is very simple. I understand that for many statements it isn't easy -- or rather extremely difficult-- to make these kinds of assessments.)

If I only talk about what I have seen, then all I have seen are images (and they keep changing too).

I disagree that it is a statement of mathematical truth. It is a re-statement of a definition.

Some math definitions of operations:
a plus a = 2 multiplied times a
a+a=2a

or:
a multiplied by a = a squared.

(a+b)^2=a^2+b^2+2ab
is simply a definition statement of the distributive and commutative laws for defined math functions.

(a+b)^2 = (a+b) * (a+b) ... that is the definition of the exponent use in math
(a+b) * (a+b) = a * (a+b) + b * (a+b) ... that is the distributive property of numbers (including real numbers)
a * (a+b) + b * (a+b) = a*a + a*b + b*a + b*b ... that is again the distributive property of numbers (including real numbers)
a*b = b*a and therefore a*b +b*a = 2ab ... associative property
a*a = a^2 and b*b = b^2 that is the definition of exponents in math

So what you have is simple algebraic use if the general properties of ALL numbers, as defined at a much lower level.

I did not watch the video. But I see no reason to even WANT to conceptualize math without the Real numbers. Losing pi would be an immediate problem for me.

 

[dkotschessaa]

I did not watch the video. But I see no reason to even WANT to conceptualize math without the Real numbers. Losing pi would be an immediate problem for me.

If I get him right (which I might not) one doesn't necessarily lose pi, but one recognizes that one can only calculate pi to a finite number of digits. The same is true of the reals. What I don't know is why he thinks that matters. Clearly we can reason about things we cannot calculate directly.

-Dave K

 

[votingmachine]

If I get him right (which I might not) one doesn't necessarily lose pi, but one recognizes that one can only calculate pi to a finite number of digits. The same is true of the reals. What I don't know is why he thinks that matters. Clearly we can reason about things we cannot calculate directly.

-Dave K

That sounds like a re-capitulation of an understanding of significant digits. If I measure a square as having sides of 1.0, I can report the diagonal as 1.4. If I measure at 1.00, I can report the diagonal as the square root of 2.00, and so on. The imprecision of my knowledge of the exactness of the sides prevents me from knowing the square root of that with any greater exactness. And if by chance the side I measured as 1.0 turns out to be 0.980 with a more precise ruler, then the diagonal is of course 1.400.

I agree that knowing significant digits does not prevent me from reasoning about an abstract perfect square with sides exactly integer "1". There is nothing wrong with saying the square-root-of-two does not exist in the rational number set. And does in the real number set. I don't grasp being "against" real numbers. I rather enjoy them.

I'm not even against complex numbers. Only Lonely numbers are bad:

One is the loneliest number that you'll ever do
Two can be as bad as one
It's the loneliest number since the number one

 

[dkotschessaa]

That sounds like a re-capitulation of an understanding of significant digits. If I measure a square as having sides of 1.0, I can report the diagonal as 1.4. If I measure at 1.00, I can report the diagonal as the square root of 2.00, and so on. The imprecision of my knowledge of the exactness of the sides prevents me from knowing the square root of that with any greater exactness. And if by chance the side I measured as 1.0 turns out to be 0.980 with a more precise ruler, then the diagonal is of course 1.400.

I agree that knowing significant digits does not prevent me from reasoning about an abstract perfect square with sides exactly integer "1". There is nothing wrong with saying the square-root-of-two does not exist in the rational number set. And does in the real number set. I don't grasp being "against" real numbers. I rather enjoy them.

I'm not even against complex numbers. Only Lonely numbers are bad:

One is also a happy number. So it's not all bad.

 

[Alan McIntire]

There are some logical problems when dealing with infinite sets or transcendental numbers. In theory, one can cut a sphere into 7 sections, and rearrange it into a larger sphere, which is intuitively absurd. Roger Penrose suggested that this was a result of the axiom of choice, which cannot be true of sets of transcendental numbers. How can one pick a number from a set of transcendental numbers which cannot even be described or named?

 

[FactChecker]

If only the real world would play along, we can limit math to exclude irrational numbers. Unfortunately, there are actual physical items like the circumference of a circle or the hypotenuse of a right triangle with unit sides. Limiting mathematics so that it can not represent those physical items with complete accuracy would be bad. It would give up the abstraction that is so important to mathematics simply because the method of measurement is more difficult for irrational numbers. That strikes me as being completely counter to the spirit of mathematics.
Furthermore, what is irrational in one system is rational in others. If I am measuring lengths along a line, there is a distance √2 when measured the usual way. Suppose I construct a right triangle with unit sides and use its hypotenuse to measure length. Then the distance √2 is 1 in the new units. Similarly, if I have a rolling wheel of diameter 1 and roll that wheel to measure distance by revolutions, I have lengths if units of π. I contend that if measuring 1 revolution exactly is possible, then measuring π exactly is possible.

 

[fresh_42]

How can one pick a number from a set of transcendental numbers which cannot even be described or named?

Easy: Pick an element from $\{\pi,e\}$. The axiom of choice starts to become problematic with uncountable infinite sets. (And I think the real problem in your example is our understanding of dimensionless points, lines or planes and not so much AC.)

 

[micromass]

There are some logical problems when dealing with infinite sets or transcendental numbers. In theory, one can cut a sphere into 7 sections, and rearrange it into a larger sphere, which is intuitively absurd. Roger Penrose suggested that this was a result of the axiom of choice, which cannot be true of sets of transcendental numbers. How can one pick a number from a set of transcendental numbers which cannot even be described or named?

Restrict your theory to measurable sets and it's solved.

 

[fresh_42]

If only the real world would play along, we can limit math to exclude irrational numbers.

The real world is discrete and finite, $\{1,\ldots,n_{max}\}$ would do.

 

[micromass]

The real world is discrete and finite, $\{1,\ldots,n_{max}\}$ would do.

Exactly. I haven't done it, but I'm 100% sure that all the math needed in physics can be done in $\{1,...,n_{max}\}$. But accepting infinite sets gives you the same results, but the theory becomes vastly simpler.

 

[bubsir]

I prefer an axiomatic approach to mathematics. Anyone who wants to introduce a "new" mathematics needs to start with their "new" axioms. I was not able to find an easy link to Mr Wildberger's axioms and therefore wouldn't suggest the work to others as anything more than entertainment.

In his introduction to "Divine Proportions" he states:
"This book revolutionizes trigonometry, re-evaluates and expands Euclidean geometry..."
I therefore assume we have Euclid's axioms including parallel lines not intersecting "to infinity!"

Shortly thereafter he begins using the notation (quantity)² without ever defining multiplication or what set of objects we might be operating on. One must make some assumption that these are "numbers." Is he assuming some other axioms as well?

 

[micromass]

I prefer an axiomatic approach to mathematics. Anyone who wants to introduce a "new" mathematics needs to start with their "new" axioms. I was not able to find an easy link to Mr Wildberger's axioms and therefore wouldn't suggest the work to others as anything more than entertainment.

In his introduction to "Divine Proportions" he states:
"This book revolutionizes trigonometry, re-evaluates and expands Euclidean geometry..."
I therefore assume we have Euclid's axioms including parallel lines not intersecting "to infinity!"

Shortly thereafter he begins using the notation (quantity)² without ever defining multiplication or what set of objects we might be operating on. One must make some assumption that these are "numbers." Is he assuming some other axioms as well?

I agree completely with you. In his YouTube video on the OP, he starts of by doing things with the parabola. But what is a parabola to him? Does it contain finitely many points, or what is it? I'm actually - for philosophical reasons - very interested in this kind of hyperfinitism. But he needs to make it logically sound with definitions and axioms. As far as I know, there is no theory of hyperfinitism that actually works and includes a good deal of mathematics. Too bad since I'm very intrigued if it would work.

 

[sophiecentaur]

I prefer an axiomatic approach to mathematics.

Yes - of course. Maths is based on axioms and, some branches of Maths happen to produce good models of the Physical World. But the result of any mathematical computation cannot automatically be said to be relevant to the real work.
None of this needs to be a problem if one adopts a grown up attitude to things and avoids looking for 'ultimate truths'. That way, you can never be disappointed or feel threatened.
I had a wonderful experience in my first year University course which prepared me, mentally for everything I later dealt with. There were a series of lectures on "Analysis", in which they started with the most basic things, like defining zero and unity and it took us, logically, into levels that I never really 'properly understood' (Sheer laziness on my part, mostly and I told myself that I was a Physicist). But it brought it home to me that Maths is 'just' a construct - but self consistent. Which doesn't mean it's true.

 

[dkotschessaa]

I agree completely with you. In his YouTube video on the OP, he starts of by doing things with the parabola. But what is a parabola to him? Does it contain finitely many points, or what is it?

I have no idea what this means or why at this point he seems to appeal to infinity, but there it is:

In affine geometry the parabola is the distinguished conic which is tangent to the line at infinity. In everyday life, the parabola occurs in reflecting mirrors and automobile head lamps, in satellite dishes and radio telescopes...
http://hrcak.srce.hr/file/169113
​

 

[sophiecentaur]

Does it contain finitely many points, or what is it?

Yes, The course I describe in the above post dealt, at length. with the importance of the terms like 'continuous' , 'differentiable' and closed and open intervals. Jumping in with 'A Parabola" is not a credible approach.

 

[micromass]

I have no idea what this means or why at this point he seems to appeal to infinity, but there it is:

In affine geometry the parabola is the distinguished conic which is tangent to the line at infinity. In everyday life, the parabola occurs in reflecting mirrors and automobile head lamps, in satellite dishes and radio telescopes...
http://hrcak.srce.hr/file/169113
​

That's just projective geometry. You can make that rigorous without appealing to infinity. But I'm still interested in a more logically ordered approach to his math.

 

[sophiecentaur]

That's just projective geometry. You can make that rigorous without appealing to infinity. But I'm still interested in a more logically ordered approach to his math.

Which is just leaping into the subject, long after all the axioms (and the hard work) has all been done. It's a bit like inventing a maths that's based on what your hand calculator tells you.

 

[micromass]

Which is just leaping into the subject, long after all the axioms (and the hard work) has all been done. It's a bit like inventing a maths that's based on what your hand calculator tells you.

Exactly. He just does the same existing math over again, but in a different (more complicated) style. It's interesting for philosophical reasons. But he'll never develop actual new math that way without doing it in R first. It's also going to be very difficult to teach in a structured way.

 

[sophiecentaur]

It's also going to be very difficult to teach in a structured way.

It should be kept well away from 'students'. It will just upset and confuse them. Sounds more like an ego trip to me.

 

[fresh_42]

There is an actual branch in mathematics that deals with those questions and as far as I can assess, with rigor and axiomatically founded (at least those who dealt with it at my university have all been logicians): https://en.wikipedia.org/wiki/Constructivism_(mathematics)

 

[dkotschessaa]

Before we beat this horse to death, here is a semi-rant by Wildberger himself:
https://njwildberger.com/2014/10/06...ion-and-my-recent-debate-with-james-franklin/

And here is the recording of a debate he had:

 

[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.