Intuisionist math and the nature of time

[Strange]

Interesting article on how using "intuitionist" mathematics might explain why time appears to flow and even unite quantum and classical theory: https://www.quantamagazine.org/does-time-really-flow-new-clues-come-from-a-century-old-approach-to-math-20200407/

Quote

Over the past year, the Swiss physicist Nicolas Gisin has published four papers that attempt to dispel the fog surrounding time in physics. As Gisin sees it, the problem all along has been mathematical. Gisin argues that time in general and the time we call the present are easily expressed in a century-old mathematical language called intuitionist mathematics, which rejects the existence of numbers with infinitely many digits. When intuitionist math is used to describe the evolution of physical systems, it makes clear, according to Gisin, that “time really passes and new information is created.” Moreover, with this formalism, the strict determinism implied by Einstein’s equations gives way to a quantum-like unpredictability. If numbers are finite and limited in their precision, then nature itself is inherently imprecise, and thus unpredictable.

 

Quote

By grounding mathematics in what can be constructed, intuitionism has far-reaching consequences for the practice of math, and for determining which statements can be deemed true. The most radical departure from standard math is that the law of excluded middle, a vaunted principle since the time of Aristotle, doesn’t hold. The law of excluded middle says that either a proposition is true, or its negation is true — a clear set of alternatives that offers a powerful mode of inference. But in Brouwer’s framework, statements about numbers might be neither true nor false at a given time, since the number’s exact value hasn’t yet revealed itself.

 

Some more background here: https://en.wikipedia.org/wiki/Intuitionism

[taeto]

What kind of object is "the fog surrounding time in physics" exactly? This all sounds like a pseudoscience scam. But in Quanta, really?

[StringJunky]

7 minutes ago, taeto said:

What kind of object is "the fog surrounding time in physics" exactly? This all sounds like a pseudoscience scam. But in Quanta, really?

It's a figure of speech, with the analogy of a 'fog' being that which obscures what we want to know. It's a metaphor for ignorance.

Edited April 9, 2020 by StringJunky

[taeto]

1 minute ago, StringJunky said:

It's a figure of speech with the analogy of a 'fog' being that which obscures what we want to know.

That being what?

[StringJunky]

1 minute ago, taeto said:

That being what?

Ignorance, probably.

[taeto]

7 minutes ago, StringJunky said:

Ignorance, probably.

So the probable meaning is that physicists are ignorant about time. Or just "ignorance surrounds time in physics", which somehow seems a more general statement. Good to know in either case. The "time" that is spoken of may or may not be the usual fourth coordinate of spacetime, or possibly neither one of those possibilities. 

[Strange]

8 hours ago, taeto said:

What kind of object is "the fog surrounding time in physics" exactly? 

It is the differences in how we perceive time and how science describes it (and the fact that different parts of physics treat it in different ways). As the article says:

Quote

Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.

In Albert Einstein’s theory of relativity, for example, time is woven together with the three dimensions of space, forming a bendy, four-dimensional space-time continuum — a “block universe” encompassing the entire past, present and future. 

Physicists who think carefully about time point to troubles posed by quantum mechanics, the laws describing the probabilistic behavior of particles. At the quantum scale, irreversible changes occur that distinguish the past from the future

I expected not to like the ideas presented in the article but, in fact, I found it a fascinating idea.

[taeto]

There are so many misunderstandings and factual mistakes in the two quotes that it is hard to know where to begin. If time is viewed as an object of philosophical study, then the word may just happen to describe something else than the version of time which in physics represents one of the coordinates of an event and which can be measured. And in that case I am not qualified to comment. But it is clear that the ideas that are presented about how mathematics works are somewhat misguided. What stands out is that whereas "intuitionist mathematics", which actually does not describe any well-defined theory or language, does do different things than usual mathematics: It is notoriously unable to do more than usual mathematics already does. It differs mainly by declaring that there are facts that cannot be shown by using their more restricted logic. In particular, if you can get something out of an intuitionist argument, you can get the exact same from a usual argument, quite possibly with less work.

Anyway,  as one example from the quotes, to suggest that intuitionists "reject the existence of numbers with infinitely many digits" is just hilarious. No, numbers do not have digits; it is the decimal representation of a real number that has digits. The theory of real numbers and their arithmetic is extremely well understood and known to be consistent and complete. It is the theory of the arithmetic of whole numbers that is known to be incomplete and not known to be consistent, presumably those are the kinds of numbers the existence of which is not rejected by intuitionists. 

Edited April 10, 2020 by taeto

[studiot]

8 hours ago, Strange said:

It is the differences in how we perceive time and how science describes it (and the fact that different parts of physics treat it in different ways).

+1

Which is why the article is way too narrow in its persective.

It is too narrow in view of mathematics being purely about numbers and number theory

It doesn't mention variables and dependence/independence

It misses the link between time and mathematics in economics, thermodynamics, chemistry (each of which offer special relationships)

 

[taeto]

The electron on the Moon versus on the Earth example at the end of the article seems suspect, as it clearly suggests that even positive numbers are zero, once they are small enough. That does not even agree with intuitionist mathematics. 

Anyway, it comes down to formalism what it all means. If limits are not defined, how is the momentum of a particle represented? Similarly how about forces and work? There is not much hint of that in the article, so I suppose it is necessary to look up the actual research papers.

Oh, and before I forget again. "Constructivist mathematics" has a different view of reality than the scientific view. By saying that an object only "exists" if it can be constructed. A zoologist should be well able to define "a cat", even without demonstrating the ability to construct a cat. In practice, similarly as in ordinary non-constructivist mathematics, it is enough to have the ability to identify whether an object that gets presented is in fact a cat or not. 

Edited April 12, 2020 by taeto

[studiot]

1 hour ago, taeto said:

Oh, and before I forget again. "Constructivist mathematics" has a different view of reality than the scientific view. By saying that an object only "exists" if it can be constructed. A zoologist should be well able to define "a cat", even without demonstrating the ability to construct a cat. In practice, similarly as in ordinary non-constructivist mathematics, it is enough to have the ability to identify whether an object that gets presented is in fact a cat or not. 

Yeah, +1

I have heard of constructivism, but not intuitionism.

Not sure of the difference or why two different words are needed.

[taeto]

39 minutes ago, studiot said:

I have heard of constructivism, but not intuitionism.

Not sure of the difference or why two different words are needed.

Wikipedia says: "Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics."

Although it does seem to me personally, from my own philosophically intuitionist point of view, that these categories are purely philosophical and have little if nothing to do with the actual practice of mathematicians. By qualifying my stance as such, I believe that I can make stuff up of any kind I wish anyway.

[MigL]

Interesting.
We view the number 'system' as continuous; we can put infinitely many numbers between any two numbers.
Could the 'solution' to our problems with time and unification be as simple as using a discrete, or 'quantized', number 'system' to model reality ?

It certainly warrants further investigation.

[dimreepr]

5 minutes ago, MigL said:

Interesting.
We view the number 'system' as continuous; we can put infinitely many numbers between any two numbers.
Could the 'solution' to our problems with time and unification be as simple as using a discrete, or 'quantized', number 'system' to model reality ?

It certainly warrants further investigation.

Saying intuition could be valid in science, seems like a dangerous president.

[taeto]

41 minutes ago, MigL said:

Interesting.
We view the number 'system' as continuous; we can put infinitely many numbers between any two numbers.
Could the 'solution' to our problems with time and unification be as simple as using a discrete, or 'quantized', number 'system' to model reality ?

It certainly warrants further investigation.

We could easily describe every geometric aspect of spacetime by using only rational numbers, or if you prefer, all numbers that can be written as rational numbers with a power of 10 in the denominator (having so-called "finite decimal representation"). Geometry does not worry about continuum, at least not in the locally Euclidean case. The problem comes with the dynamics; a velocity is defined by differentiation, and derivatives of rational-valued functions are not rational-valued. 

Edited April 12, 2020 by taeto

[studiot]

3 hours ago, taeto said:

We could easily describe every geometric aspect of spacetime by using only rational numbers, or if you prefer, all numbers that can be written as rational numbers with a power of 10 in the denominator (having so-called "finite decimal representation"). Geometry does not worry about continuum, at least not in the locally Euclidean case. The problem comes with the dynamics; a velocity is defined by differentiation, and derivatives of rational-valued functions are not rational-valued. 

How does Pi work with this or surds??

Edited April 12, 2020 by studiot

[taeto]

6 minutes ago, studiot said:

How does Pi work with this or surds??

You mean geometrically? For \(\pi\) I suppose you would be able to show that all circles of equal radius are congruent. Classically you do not have any immediate notion of measures or lengths, other than one length can be half of another one, or three times as long. When you later introduce length, you define the ratio between circumference and diameter to be \(\pi\). That does not require any further investigation into the nature of the number \(\pi;\) for what we know from this definition, it could be rational or not, supposing we have such a notion. Similarly with the length of the hypotenuse of a right triangle with two other congruent sides. The notion of "lengths" here are really quite separate from the specifics of the kinds of points that make up the line segments and arcs. Even line pieces that consist of only "rational points" are congruent to and have the same lengths as the equivalent line pieces in which all missing "real points" are added.  

[studiot]

1 hour ago, taeto said:

You mean geometrically? For \pi I suppose you would be able to show that all circles of equal radius are congruent. Classically you do not have any immediate notion of measures or lengths, other than one length can be half of another one, or three times as long. When you later introduce length, you define the ratio between circumference and diameter to be \pi . That does not require any further investigation into the nature of the number \pi; for what we know from this definition, it could be rational or not, supposing we have such a notion. Similarly with the length of the hypotenuse of a right triangle with two other congruent sides. The notion of "lengths" here are really quite separate from the specifics of the kinds of points that make up the line segments and arcs. Even line pieces that consist of only "rational points" are congruent to and have the same lengths as the equivalent line pieces in which all missing "real points" are added.  

And angle ?

[taeto]

1 hour ago, studiot said:

And angle ?

Classically an angle is made up of two rays (half-lines) that meet in a common endpoint. If you want to measure an angle, you can compare it to one of the arcs of a unit circle that gets cut by placing the meeting point of the two rays in the center of the circle. Then the length of this arc is some fraction of the length \(2\pi\) of the arc of the whole circle, and that determines the measure of the angle relative to \(2\pi.\) Again it is immaterial whether \(\pi\) is a whole number, a fraction of a whole number, or irrational.  

Edited April 12, 2020 by taeto

[studiot]

But 2pi is irrational.

So 1 radian is also irrational.

Your thesis was you can do geometry with only rational numbers.

But you can't if you have to make a fraction with an irrational one.

 

 

Edited April 12, 2020 by studiot

[taeto]

19 minutes ago, studiot said:

But 2pi is irrational.

So 1 radian is also irrational.

Your thesis was you can do geometry with only rational numbers.

But you can't if you have to make a fraction with an irrational one.

No, you do not do geometry with numbers, that is the point. You may in Euclidean Geometry have a picture of lines being composed of points that have rational coordinates, or of algebraic coordinates, or coordinates that are constructible numbers, it does not matter. The concept of a number being rational or not does not really enter into it. You have to think about geometry in terms of geometrical objects being parallel, congruent, similar, etc. all of which do not involve numbers. Actually, just think about what can be done if you do not have any \(x\)- or \(y\)-axes. Geometry is about the relationships that you can still identify and prove without those coordinates.

Just go through Euclid's axioms for the Euclidean plane, and observe that it does not matter when you talk about points and lines whether the points are "rational" or not, in fact there is no way to even speak of this. Until you introduce added-on axioms like the Cantor-Dedekind axiom, which talks about limits. But that came in a bit later after the original axioms. 

Edited April 12, 2020 by taeto

[studiot]

10 minutes ago, taeto said:

No, you do not do geometry with numbers, that is the point. You may in Euclidean Geometry have a picture of lines being composed of points that have rational coordinates, or of algebraic coordinates, or coordinates that are constructible numbers, it does not matter. The concept of a number being rational or not does not really enter into it. You have to think about geometry in terms of geometrical objects being parallel, congruent, similar, etc. all of which do not involve numbers. Actually, just think about what can be done if you do not have any x - or y -axes. Geometry is about the relationships that you can still identify and prove without those coordinates.

Just go through Euclid's axioms for the Euclidean plane, and observe that it does not matter when you talk about points and lines whether the points are "rational" or not, in fact there is no way to even speak of this. Until you introduce added-on axioms like the Cantor-Dedekind axiom, which talks about limits. But that came in a bit later after the original axioms. 

Perhaps we have read different history (of Mathematics) books ?

There is no coordinate Geometry in Euclid, that was introduced by Descartes and is therefore also called Cartesian Geometry.

But Euclidian Geometry does have a concept of length, you cannot have congruence without it, only similarity (which is also included in Geometry).

Much of Euclidian Geometry is about congruence and other measures such as Pythagoras.

Indeed the Ancient Greeks who proved that √2 is irrational nearly had a war over it.

But you can define Geometry any way you want. You just have to try to persuade others to use your definition if it differs from convention.
And of course convention changes (some say develops). The first half of the 20th century was occupied by those who wnated to show that all Geometry is algebraic in nature.
On the other hand the Ancient Greeks were more constructivist in that they even had a definition of what you could use to 'construct' a figure.
But they could not construct a Mobius strip with their methods.

So if I want to use the full range of angles or linear (or area or volume) measures  I am forced to accept irrational (and other) numbers.

[taeto]

5 minutes ago, studiot said:

Perhaps we have read different history (of Mathematics) books ?

There is no coordinate Geometry in Euclid, that was introduced by Descartes and is therefore also called Cartesian Geometry.

But Euclidian Geometry does have a concept of length, you cannot have congruence without it, only similarity (which is also included in Geometry).

Much of Euclidian Geometry is about congruence and other measures such as Pythagoras.

Indeed the Ancient Greeks who proved that √2 is irrational nearly had a war over it.

But you can define Geometry any way you want. You just have to try to persuade others to use your definition if it differs from convention.
And of course convention changes (some say develops). The first half of the 20th century was occupied by those who wnated to show that all Geometry is algebraic in nature.
On the other hand the Ancient Greeks were more constructivist in that they even had a definition of what you could use to 'construct' a figure.
But they could not construct a Mobius strip with their methods.

So if I want to use the full range of angles or linear (or area or volume) measures  I am forced to accept irrational (and other) numbers.

Yes, you can slide in Cartesian Geometry sideways if you want to. But the point of this thread was the role of the coordinate \(t\) for time in physics. And my point is then that it cannot really matter which kind of numbers we choose as the range of \(t\) geometrically, because that is not relevant for the kind of geometry that we posit as a model for physical space, which is Euclidean. Now you take an intuitionist view to say that we may define geometry any way we want. But if we talk about Euclidean Geometry, there are some ground rules.

Actually, congruence in Euclidean Geometry does not involve "length". It only involves whether it is possible to move one object exactly on top of another object by using certain allowed operations. There is nothing about measurements and numbers.

[studiot]

12 hours ago, taeto said:

Yes, you can slide in Cartesian Geometry sideways if you want to. But the point of this thread was the role of the coordinate t for time in physics. And my point is then that it cannot really matter which kind of numbers we choose as the range of t geometrically, because that is not relevant for the kind of geometry that we posit as a model for physical space, which is Euclidean. Now you take an intuitionist view to say that we may define geometry any way we want. But if we talk about Euclidean Geometry, there are some ground rules.

Actually, congruence in Euclidean Geometry does not involve "length". It only involves whether it is possible to move one object exactly on top of another object by using certain allowed operations. There is nothing about measurements and numbers.

I didn't want to introduce Cartesian Geometry by any particular door, but Euclidian Geometry is not a coordinate Geometry.

Euclidian Geometry does require the concept of coordinate independent length as a magnitude and angle as a magnitude.

Definitions, 9,10 11, 12 and 15

Axioms 3,4 and 5

Postulates 1,2,3 and 4

all depend upon this.

Further in Euclidian Geometry it is not necessary to superpose figures to establish congruence.
Using the numeric  proporties of the lines and angles as stated suffices.

Superposition adds an extra axiom that 'moving' the figure does not distort these magnitudes.

 

This lack of a need for any coordinate system is also implicit in Relativity (enter the Physics) .

 

Another aspect is that constructivism requires a causal ordering for many propositions that flow from the definitions and axioms, this also is preserved in Special Relativity.

 

Finally there is the fact that (as Strange noted) different parts of Physics view time in different ways,

I extended that beyond Physics in my first post in this thread.
No one seems to have taken that up.

 

[taeto]

9 minutes ago, studiot said:

Euclidian Geometry does require the concept of coordinate independent length as a magnitude and angle as a magnitude.

Definitions, 9,10 11, 12 and 15

Axioms 3,4 and 5

Postulates 1,2,3 and 4

all depend upon this.

Indeed. Coordinate independent means that it does not depend on the coordinates what is meant by measures of length and angle. I can say this with some confidence, since if we were to observe a geometric object all of whose points are represented by rational coordinates, then by shifting the coordinates by \(\sqrt{2}\) we obtain just another representation of the exact same object, but using all irrational coordinates. Shift by \(\pi\) and you get transcendent coordinates. 

Also, it is the measure of an angle which is a magnitude. The angle itself is a geometric figure.

18 minutes ago, studiot said:

Further in Euclidian Geometry it is not necessary to superpose figures to establish congruence.
Using the numeric  proporties of the lines and angles as stated suffices.

Numeric values may suffice, once you have the necessary additional notions in place. And this will need proof. But the definitions are not based on having numerical measures. 

24 minutes ago, studiot said:

Another aspect is that constructivism requires a causal ordering for many propositions that flow from the definitions and axioms, this also is preserved in Special Relativity.

I do not know what you mean by this. But I could get nasty (towards the constructivists) by suggesting that \(c,\) the speed of light, which features prominently in SR and GR, may just happen to not have any representation in terms of a quantity which is not rejected by constructivists.  

27 minutes ago, studiot said:

Finally there is the fact that (as Strange noted) different parts of Physics view time in different ways,

I extended that beyond Physics in my first post in this thread.
No one seems to have taken that up.

I always think of time in physics as one of the coordinates by which you describe the location of an event. And usually the range of this coordinate is real numbers, because that is just extremely convenient for talking about limits, such as when differentiating or integrating functions of time to get theoretical values of physical quantities. But it should not really matter which real numbers you use for any static description; you might use rational numbers for specifying positions of physical objects, or even whole numbers, if your choice of unit makes it small enough. And this is what I think of as the classical geometric view. As I mentioned, if you have any system given by coordinates in which all positions are represented by whole numbers, you may shift coordinates to represent the same system by non-constructible transcendental numbers. That being one reason why I cannot make any sense of the article that is referred to in the OP.

I take up one view of time which is employed by constructivists in particular. They would say, I believe, that if you investigate a number with a decimal representation like \(0.4999999...\) then it will take you time to discover whether the number is actually equal to \(1/2.\) And to ascertain that it is actually equal to \(1/2\) will take you infinite time. It seems that the new explanation of time that gets offered is dependent on using numbers that require only finite time to represent. So that there is one view of time that is used to explain a different view of time.