Physics Lournal

Classical and Intuitionist Mathematics Shapes Our Understanding of Time

in Physics

Metadata

Author:: Nicolas Gisin

Abstract:

Physics is represented in terms of timeless classical mathematics. Presenting this in terms of Intuitionist Mathematics, built on processes of time evolution, offer a different perspective that maps more closely onto physical reality.

In 1922, Albert Einstein, met up with a philosopher, Henri Bergson, where they debated about time, wherein Einstein stated that: "There is no such thing as the time of the philosopher."

Around this time, mathematicians were in the middle of an equally important discussion, about how to properly describe a continuum.

The mathematician David Hilbert was promoting a formalized mathematics, that treats a number with an infinite amount of digits as a real physical object.

Opposite him, was Dutch mathematician LEJ Brouwer, supporting the view that the line should be seen as a never-ending process, that develops through time, the core of intuitionist mathematics.

Brouwer was supported by people like Weyl, and Gödel, but Hilbert's formulation came out on top, leading to the expulsion of time from mathematics, and mathematical objects were thus seen as objects that exist in some platonic realm.

These debates impacted Physics greatly: mathematics is the language of physics, and platonism makes speaking about time challenging. This led to the expulsion of the concept of time in physics: all events are the consequence of quantum fluctuations that took place during the big bang.

Subsequently, in todays physics, there is no creative time, and there is no "now".

The consequence of this was dramatic, when considering physics doesn't just concern technologies and abstract theories, but also the fundamental workings of nature.

Time is an integral part of the human experience, as pointed out by Yuval Dolev: "To think of an event is to think of something in time. [...] Tense and passage are not removable from how we think about events.

From this it may seem that physics should avoid narrating the human experience and stick to physical theory, but this may be dangerous to physics, and how well it is capable of responding to phrases such as "Time is an illusion".

Science without time is only ruin of intelligibility. - Rabelais

Outside of these debates, it is argued in Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?, that a finite amount of space cannot logically be said to contain more than a finite amount of information.

The mathematics used by scientists makes formulating concepts such as the passage of time difficult.

Bergson never accepted Einstein's statement, and Einstein himself was unsettled by the fact that physics lacked a proper description of the concept of now, though he himself was not sure of how to fold the concept in. Hilbert as well, was made uncomfortable by the infinities associated with his mathematics, stating that physics should never incorporate actual infinities.

In order to demonstrate the tension faced by physics, we can observe Classical Mechanics.

Classical mechanics is usually represented by clocks and cogs, but most classical dynamical systems differ from these because they are chaotic, due to the hypersensitivity of the evolution of the system to the initial conditions of the system, the future of classical systems depends on digits very far into the sequence of some value of a parameter describing the system.

A simple example of these systems can be modeled like so:

Assume the state-space reduces to the unit interval, and the system is stroboscopic.

The state is thus described by a number comprised of an infinity of digits that follow zero, and at each time step, the digits are shifted left, with the first digit being discarded:

$X_{t} = 0.b_1b_2b_3b_4 \ldots b_n \ldots$

$X_{t + 1} = 0.b_2b_3b_4b_5 \ldots b_n \ldots$

After n time-steps, the nth digit is the most relevant digit for (accurately) representing the state of the system.

If one assumes all of the digits of some initial condition represent physical reality, of some system, such as the weather, has an entirely predetermined course of behavior.

This displays the absence of creative time: nothing really happens, but rather everything is a result of the initial conditions and the deterministic evolution of the system: this is a consequence of representing the infinite number of points on a line between 0 and 1 in classical platonist mathematics.

The question now is, is the above representation necessary?

Most physicists seem to reject such a representation, but also admit that they don't see another way of going about their work.

Max Born, one of the fathers of Quantum Mechanics, stated "...Statements like a quantity x has a completely definite value (expressed by a real number and represented by a point in the mathematical continuum) seem to me to have no physical meaning ".

Intuitionist mathematics can be of use in this situation.

In this school of mathematics, numbers are processes that develop over time: at each moment of time there is only a finite amount of information.

To demonstrate this, we can consider a quantum RNG, not as a human creation, but as a a result of the way nature works: intrinsically and fundamentally indeterministic.

This source of randomness feeds into the digits of real numbers as illustrated in the following figure:

One way to understand the continuum in the context of intuitionistic mathematics, is to assume that nature is capable of generating true randomness, that outputs a digit $r(n)$ , at each time step $n$ . Furthermore, at each time step, a rational number $\alpha(n)$ , is computing by a function $\small f: \alpha(n) = f(\alpha(n - 1), n, r(1), \ldots), r(n)$ . Different functions $f$ represent different classes of series of $\alpha(n)$ . This series $\alpha(n)$ is assumed to converge, though at any given time step, only finite amounts of information about the series exists, according to the idea that the RNG is a truly endless process that develops in time.

A simple example assumes that the function $f$ merely adds the random digit $r(n)$ as the nth digit of $\alpha$ :

$\alpha(n) = \alpha(n-1) + r(n) \cdot 10^{-n} =$

$0.r(1)r(2)r(3) \ldots r(n)$

Often it is assumed that $\alpha(0)$ is assigned a initial rational number, but this is not essential. Also, if the RNG is actually a pseudo-random number generator, then $\alpha(n)$ converges to a computable number.

In other examples the digits of $\alpha(n)$ are correlated, meaning new digits of $\alpha(n)$ depend on the previous $k$ numbers $r(j)$ . By selecting the appropriate functions $f$ , an infinity of classes of series can be defined. Historically, Brouwer, did not use RNG's, but mathematical objects he called choice sequences, where the choices were made by an idealized mathematician, and what are referred here to as classes, were termed spreads by Brouwer.

It should be noted that, typically, real numbers are truly random, just as are the outcomes of quantum measurements, as has been pointed out by Gregory Chaitin. Additionally, real numbers tend to contain infinite amounts of information, allowing for the encoding in a single number, the answers top all the questions that can be formulated in human language, as pointed out by Emile Borel.

This leaves us with a choice: either we accept that all of the digits of some initial conditions are are determined from the first moment, resulting in timeless physics, or we accept that these digits are indeterminate initially, and physics thus contains events that take place as time passes. Both cases allow for chaotic systems to exhibit randomness.

In the first case, from the Platonistic perspective, all randomness is encoded within initial conditions.

In the second case, randomness emerges as time passes, as is described by the intuitionist perspective. Note that intuitionism doesn't derive determinism, but assumes it from the start, as opposed to classical mathematics, which assumes infinite information in the initial conditions.

Referenced in

Nicolas Gisin

[[Classical and Intuitionist Mathematics Shapes Our Understanding of Time

in Physics]]

Academic Papers on Mathematics

[[Classical and Intuitionist Mathematics Shapes Our Understanding of Time

in Physics]]

Classical and Intuitionist Mathematics Shapes Our Understanding of Time in Physics

Classical and Intuitionist Mathematics Shapes Our Understanding of Time

Metadata

Author:: Nicolas Gisin

Abstract:

Physics is represented in terms of timeless classical mathematics. Presenting this in terms of Intuitionist Mathematics, built on processes of time evolution, offer a different perspective that maps more closely onto physical reality.

In 1922, Albert Einstein, met up with a philosopher, Henri Bergson, where they debated about time, wherein Einstein stated that: "There is no such thing as the time of the philosopher."

Around this time, mathematicians were in the middle of an equally important discussion, about how to properly describe a continuum.

The mathematician David Hilbert was promoting a formalized mathematics, that treats a number with an infinite amount of digits as a real physical object.

Opposite him, was Dutch mathematician LEJ Brouwer, supporting the view that the line should be seen as a never-ending process, that develops through time, the core of intuitionist mathematics.

Brouwer was supported by people like Weyl, and Gödel, but Hilbert's formulation came out on top, leading to the expulsion of time from mathematics, and mathematical objects were thus seen as objects that exist in some platonic realm.

These debates impacted Physics greatly: mathematics is the language of physics, and platonism makes speaking about time challenging. This led to the expulsion of the concept of time in physics: all events are the consequence of quantum fluctuations that took place during the big bang.

Subsequently, in todays physics, there is no creative time, and there is no "now".

The consequence of this was dramatic, when considering physics doesn't just concern technologies and abstract theories, but also the fundamental workings of nature.

Time is an integral part of the human experience, as pointed out by Yuval Dolev: "To think of an event is to think of something in time. [...] Tense and passage are not removable from how we think about events.

From this it may seem that physics should avoid narrating the human experience and stick to physical theory, but this may be dangerous to physics, and how well it is capable of responding to phrases such as "Time is an illusion".

Outside of these debates, it is argued in Indeterminism in Physics, Classical Chaos and Bohmian Mechanics. Are Real Numbers Really Real?, that a finite amount of space cannot logically be said to contain more than a finite amount of information.

The mathematics used by scientists makes formulating concepts such as the passage of time difficult.

In order to demonstrate the tension faced by physics, we can observe Classical Mechanics.

A simple example of these systems can be modeled like so:

Assume the state-space reduces to the unit interval, and the system is stroboscopic.

The state is thus described by a number comprised of an infinity of digits that follow zero, and at each time step, the digits are shifted left, with the first digit being discarded:

Xt=0.b1b2b3b4…bn…X_{t} = 0.b_1b_2b_3b_4 \ldots b_n \ldotsXt​=0.b1​b2​b3​b4​…bn​…

Xt+1=0.b2b3b4b5…bn…X_{t + 1} = 0.b_2b_3b_4b_5 \ldots b_n \ldotsXt+1​=0.b2​b3​b4​b5​…bn​…

After n time-steps, the nth digit is the most relevant digit for (accurately) representing the state of the system.

If one assumes all of the digits of some initial condition represent physical reality, of some system, such as the weather, has an entirely predetermined course of behavior.

The question now is, is the above representation necessary?

Most physicists seem to reject such a representation, but also admit that they don't see another way of going about their work.

Max Born, one of the fathers of Quantum Mechanics, stated "...Statements like a quantity x has a completely definite value (expressed by a real number and represented by a point in the mathematical continuum) seem to me to have no physical meaning ".

Intuitionist mathematics can be of use in this situation.

In this school of mathematics, numbers are processes that develop over time: at each moment of time there is only a finite amount of information.

To demonstrate this, we can consider a quantum RNG, not as a human creation, but as a a result of the way nature works: intrinsically and fundamentally indeterministic.

This source of randomness feeds into the digits of real numbers as illustrated in the following figure:

A simple example assumes that the function fff merely adds the random digit r(n)r(n)r(n) as the nth digit of α\alphaα:

α(n)=α(n−1)+r(n)⋅10−n=\alpha(n) = \alpha(n-1) + r(n) \cdot 10^{-n} = α(n)=α(n−1)+r(n)⋅10−n=

0.r(1)r(2)r(3)…r(n)0.r(1)r(2)r(3) \ldots r(n)0.r(1)r(2)r(3)…r(n)

Often it is assumed that α(0)\alpha(0)α(0) is assigned a initial rational number, but this is not essential. Also, if the RNG is actually a pseudo-random number generator, then α(n)\alpha(n)α(n) converges to a computable number.

In the first case, from the Platonistic perspective, all randomness is encoded within initial conditions.

In the second case, randomness emerges as time passes, as is described by the intuitionist perspective. Note that intuitionism doesn't derive determinism, but assumes it from the start, as opposed to classical mathematics, which assumes infinite information in the initial conditions.

Referenced in

Nicolas Gisin

[[Classical and Intuitionist Mathematics Shapes Our Understanding of Time

Academic Papers on Mathematics

[[Classical and Intuitionist Mathematics Shapes Our Understanding of Time

$X_{t} = 0.b_1b_2b_3b_4 \ldots b_n \ldots$

$X_{t + 1} = 0.b_2b_3b_4b_5 \ldots b_n \ldots$

A simple example assumes that the function $f$ merely adds the random digit $r(n)$ as the nth digit of $\alpha$ :

$\alpha(n) = \alpha(n-1) + r(n) \cdot 10^{-n} =$

$0.r(1)r(2)r(3) \ldots r(n)$

Often it is assumed that $\alpha(0)$ is assigned a initial rational number, but this is not essential. Also, if the RNG is actually a pseudo-random number generator, then $\alpha(n)$ converges to a computable number.