## Determinism and mathematical truth.

• 79
Extreme determinsts maintain that the physical universe is just an outworking of the laws of nature and everything is predetermined by these laws. It is even argued that our brains are determined by neurological processes etc that are physically deterministic.

There seems to be a way around this determinism. It involves making a list of possible actions and making a choice from that list in such a way that the choice is not determined by either neurological states or any physical state in the world.

Here is how it works. Make a list of ordinary events and label them 0 to 9.

1. Go to the library
2. Play tennis
4. Go to the cinema
5. Go to the supermarket
7. ...
8. ...
9. ...

Next get the decimal expansion of an irrational number such as the square root of 11
or 1/23.

Let us take the square root of 11

The square root of 11 is 3.3166247903554

Now take the first digit in the decimal expansion, 3

the next is 1

1 = Go to the library

6 = Listen to the radio

etc.

Now our choice is not determined by any physical or neurological state. It is determined by purely non physical mathematical entities. So we seem to have broken with any previously determinism by letting digits make our choice for us. If we are in the library, for example, we are engaged with a series of physical acitivities that, as a set, cannot be traced back to any previous physical state because the digit intervened and determined what set of physical events we would enter into. Comments?
• 520
The choice to follow the digits is determined, so the paradigm is still in effect. Its pretty inescapable.
• 7.7k
What DingoJones said (assuming determinism that is, not that I'm a determinist).

Also:

non physical mathematical entities.

There's no such thing as a nonphysical anything.
• 79
The choice to follow the digits is determined, so the paradigm is still in effect. Its pretty inescapable.
Within the experiment many things are arguably determined including the choice to follow the digits but the choice itself is determined by the digit, not by a brain state or physical state: the digits determine what happens next and in this sense the choice itself is not physically deterministic. The digits are not determined to be what they are by any physical state. They are eternal truths.
• 520

At each step of your process there is determism. When you’re choosing the square root of 11, what to put on your list etc etc
The “choice itself” is an illusion. It is the result of determinate factors, not some sort of causal vacuum. You are no more making a choice when those numbers play out than you are when you “choose” to continue falling after jumping off a cliff.
• 674
Now our choice is not determined by any physical or neurological state.

I like the creativity of your post. It's a fun idea.

As I think others implied, we would still have to address the decision to 'run' your algorithm in the first place. Someone could claim to ideally be able explain it in terms of physical brain stuff --but this open a supreme can of worms. What is explanation? Have I explained something just because I found an algorithm that makes good predictions? Anyway, I don't think the strong determinist (who might not even appeal to physics) would be forced to admit defeat.
• 79
At each step of your process there is determinism. When you’re choosing the square root of 11, what to put on your list etc etc

As I think others implied, we would still have to address the decision to 'run' your algorithm in the first place.

Yes, I don't assert a complete escape from determinism. There are many threads of determinism throughout the experiment. Body temperature, heart rate etc. are not necessarily altered. But it is only necessary to focus on one thread; the decision itself. The decision is made by a digit, not by a physical state, therefore it is mathematically determined, but not determined by any physical state during the experiment. Yes, I do choose the starting number, 11, but I don't decide what the digits in the decimal expansion are, nor are they physically determined, they are eternal truths and they cause the choice. It helps if I don't know what the digits are before the experiment, - that way 'brain states' cannot be invoked.

There is no mental choice per se. The choice is mathematically determined. It is determined by digits that are outside any physically deterministic thread. That is the whole point.
• 982
I think the number has to be transcendental, not algebraic and hence not just irrational. Then it Works, I assume. You see algebraic numbers are countable. The square root of two is irrational, yet it is a solution of the polynomial equation x2 − 2 = 0.

What you are using in a modified way is called Cantor's diagonalization method. Yes, it refutes this kind of simplistic determinism solely based on math and logic.

Basically it's negative self reference, which is totally possible. And it's in the root of how Gödel's Incompleteness Theorems or the argument of a Turing Machine in the Entscheidungsbproblem are made.

I think it does refute determinism in the Laplacian way, and the fact is that we simply are part of the universe. (Laplacian determinism usually is defined with the example of Laplace's demon: if someone, the Demon, knows the precise location and momentum of every atom in the universe, their past and future values for any given time are entailed; they can be calculated from the laws of classical mechanics.)

Let me give you a totally different kind of example:
You can simply then ask the Demon a simple question:

"Which number, 1 or 2, I will pick next?"
You then base your choice the following way,
If the Demon says 1, you pick 2
If the Demon says 2, you pick 1
If the Demon says anything else or stays silent, you pick 1.

Hence there is a correct forecast of what you pick, but for the Demon it should say the number that it doesn't say as you base your decision on what the Demon forcasts. That's the negative self reference. You simply cannot say something you don't say.

Now the transcendental number in decimal expansion is an endless number line that doesn't repeat itself (if it would, it wouldn't be irrational). As it doesn't repeat itself in any length, there's no way to compress it. And this "compression" basically makes it possible to have the idea of Laplacian determinism.
• 79
I think the number has to be transcendental,ssu

No. All that is required is that the digits are unknown in advance (to counter the argument for brain states making the choice.) I can say 'I will choose the 75th digit in the expansion of the square root of 7'. I then go and see what it is and act accordingly. In this way physical determinism has not made the choice, it has been replaced by mathematical determinism. Any physical determinism that would have made the choice is terminated at the point when the digit intervenes and makes the choice.
• 302
Any physical determinism that would have made the choice is terminated at the point when the digit intervenes and makes the choice.

The choice to set up choices to be followed by this procedure is the only "free" one in the system.
Strictly speaking, a view of all events being predetermined concerns whether everything is caused or not and how those causes relate to each other. It doesn't distinguish between the mathematical and the physical, or at least by itself alone.
• 762
I don't even think it's a challenge to determinism at all. It's only non-deterministic if you ignore the setup of the entire list and why you select certain actions. If you want you can refer to this as "meta-determinism" but that's going to be relevant to any discussion of determinism vs indeterminism, so this doesn't seem terribly enlightening. Like take this bit:

Now our choice is not determined by any physical or neurological state. It is determined by purely non physical mathematical entities.

No it isn't, it's determined by the marks on the paper/whatever medium (or mental representation) and what you "chose" to assign as the course of action for that particular digit. There's absolutely nothing here that invalidates any sort of determinism, even the simplistic variety. The "choice" here to set the arbitrary act to some number coming out of a random irrational number you selected can easily have been determined and so the rest is as well. You're mistaking the epistemic issue of not knowing beforehand which course of action you would take with the metaphysical issue of it being deterministic or not.
• 7k
it's determined by the marks on the paper/whatever medium (or mental representation) and what you "chose" to assign as the course of action for that particular digit.

We determine what symbols mean. They have no intrinsic meaning. That is why they can't be physical - because a physical mark is not capable of meaning anything without an interpretive act by an observing intelligence.

Nowadays people just assume that it is understood how the brain gives rise to thought and reason, but I say it's not understood at all. And that is because even in order to make any statement about what the brain does, requires rational inference, and rational inference is based purely on the relationships of ideas.

Extreme determinsts maintain that the physical universe is just an outworking of the laws of nature and everything is predetermined by these laws.

The problem with that is since the early part of the 20th Century it has become clear that the laws of nature are not themselves deterministic, but probabilistic.
• 762
We determine what symbols mean. They have no intrinsic meaning.

I don't think this is really relevant since nothing I said was about meaning and symbols, just the setup of "choosing" based on the symbol that happens to come up. I certainly never said symbols have intrinsic meaning. The interpretation of some symbol as having some meaning can itself be deterministic in which case it just further entombs the determinism.

Not that I'm claiming determinism is the case, just that nothing in the OP's exercise challenges determinism in any way.
• 982
No. All that is required is that the digits are unknown in advance (to counter the argument for brain states making the choice.) I can say 'I will choose the 75th digit in the expansion of the square root of 7'. I then go and see what it is and act accordingly. In this way physical determinism has not made the choice, it has been replaced by mathematical determinism. Any physical determinism that would have made the choice is terminated at the point when the digit intervenes and makes the choice.
So why then irrational numbers in the first place? Unknown in advance is quite a loose definition the way you say it.

Now a rational number that repeats 123123123... might be easy to remember (even easier is the one which repeats 0123456789012345...) and easy to think an algorithm that gives what's the 75th number will be. So, if I follow your reasoning (?) this then isn't unknown if I can count what the number will be.

Well, if I've understood you correctly, take a rational number that in the decimal expansion starts repeating itself only after 10 000 digits, that may then be "unknown" as it obviously is hard to find a person that remembers a 10 000 digit decimal sequence.

To really get digits to be unknown in advance and you would have to use numbers that don't have an algorithm to crunch them. That's just my point with transcendental numbers.
• 520

In what way is the “choice” the numbers make not moored to the equation itself and thus determined in precisely the same way as other deterministic processes? The outcome will always be based on a causal chain, whether the person initiating the process knows the outcome or not. There are weak points to determinism where free will is concerned but I don’t think this is one of them, its covered by determinism.
• 44
here's no such thing as a nonphysical anything.

there could be. what if we are living in a simulation?
• 7.7k

Simulations are physical.
• 79
In what way is the “choice” the numbers make not moored to the equation itself and thus determined in precisely the same way as other deterministic processes?

This is the crux of the matter. When I say 'determinism' I am talking about physical determinism which says that every physical state is determined by the previous physical state. I am replacing physical determinism with mathematical determinism. What matters is the value of the digit. This value is not physically determined, it is an eternal mathematical truth. This means that the choice is not physically determined, it is mathematically determined. The determinist would say that a choice is determined by a brain state but there is no mental choice here as it is determined not by a brain state but by a digit. The point at which determinism is broken is when the digit is correlated with a number on the list. This correlation is not physically determined as the value of the digit is not physically determined.

So why then irrational numbers in the first place?ssu

I chose irrational expansions simply because they are more in the spirit of the experiment, as opposed to predictable, repeating, rational expansions that could be known in advance. But there are all kinds of mathematical entities that would suffice.
• 79
Let me put it this way. We must be careful to keep in mind what physical determinism means. It means that any state can, in principle, be shown to be physically caused by the previous physical state. This assertion depends completely on using the physical laws of nature to demonstrate how one physical state leads to another.

Now, this physical determinism can be shown to obtain right up to the choice of the initial number (11 in this case) but if physical determinism makes the choice from the list then it must be shown how the laws of nature determine that choice. How could this happen?

It cannot happen by way of any brain state because no mental choice is made. The digit determines the choice. But if physical determinism is to obtain all the way through it must be shown how the value of the digit is determined by physical laws. But it is not. It is mathematically determined. This means that the choice is not determined by any physical law because no physical law determines the value of the digit.

Yes, there is, arguably, a great amount of determinism leading up to the arrival of the digit but there is no way that physical law can be shown to effect the choice.

The laws of nature may influence the digit that comes up but influence is not enough. It has to be shown that the choice is physically determined according to physical law but physical law cannot determine the value of the digit. So any physical determinism ends with the arrival of the digit and mathematical determinism takes over from there.

One has to be careful to distinguish between physical determinism influencing things and rigidly determining what happens according to physical law.
• 520

There is no such distinction in Determinism. Its all happening within the physical realm, the way the mind works, the way logic is informed, the basis fir rationality etc etc, its all bound together by causality and thus falls under Determinism, “physical” Determinism is a distinction you yourself are making to service your argument, but it ignores the other foundations of Determinism. Specifically, you don’t have an answer for the causal chain within which your equation exists, and I dont see how you can question Determinism without directly addressing that fundamental part of Determinism. Restricting Determinism to direct physical objects is a straw man.
• 171
Just for the record, math is a game that we play, with definitions and rules.

Math does not really exist outside of the human mind. It is simply an analytical tool that humans use to do things in an organized manner, such as count buffalo by prehistoric hunters.
• 982
I chose irrational expansions simply because they are more in the spirit of the experiment, as opposed to predictable, repeating, rational expansions that could be known in advance. But there are all kinds of mathematical entities that would suffice.
So basically irrationality or what the number would be used here wasn't important. "Known in advance" is quite vague definition here. By whom? Someone who isn't good at math (then even a rational number makes it) or the a math-enthusiast who can use his brain as a calculator?

But if physical determinism is to obtain all the way through it must be shown how the value of the digit is determined by physical laws. But it is not. It is mathematically determined.
This is a bit confusing. How do you define these two to being "physical", yet then something being "mathematical" as opposition to the first?
• 132
Those are still determinate events since you could say it was your neurological (clasical) state which chose the irrational number.
I get what you're saying though. Maybe if it was a random number gernerator it would break the chain, or something that is known to be schotastic like QM or radiation.
• 762
Just for the record, math is a game that we play, with definitions and rules.

Math does not really exist outside of the human mind. It is simply an analytical tool that humans use to do things in an organized manner, such as count buffalo by prehistoric hunters.
hks

I don't really think this is quite enough. Math is not just a game with rules and definitions. Universes have structure, structures which can be mapped onto mathematical systems whose properties we can explore. That gives a sense of "realness" to some areas of mathematics which puts it well over this seeming notion of total instrumentalism. It does not simply exist in the mind, and really, it seems to be exactly the opposite. Mathematical systems have properties the human mind does not (infinite properties, for instance).
• 171
"... a sense of realness" is NOT reality.

Reality follows patters which we can infer through scientific observation and measurement.

However the math that we humans have invented to approximate it is not reality.

Is that enough yet?
• 318
Mathematics are made by Man, but not out of the blue, but as a response to stimuli from "outside". Phenomena happen, we react, and mathematics is a tool we use when we react to phenomena. But to that tool to be useful, a real connection to real physical patterns must be present. We invented the Pithagoras´theorem, but we did not invented the proportions among the sides and angles of a triangle. Crows use Archimedes´ theorem; the formula is not verbalized in their brains, but it is codified nonetheless or they wouldn´t get the walnut in experiments. Becouse volume added to a container with liquid makes the level rise proportionally, and that´s a real mathematical law, that exists alongside our understanding of the pattern.
• 318
I´m a determinist, but I don´t like to have absolute certainties about things. So I´d really love to hear solid arguments in favour of non determinist universes. I appreciate the attempt of En Passant to provide one, even if it isn´t quite there yet.
• 762
No, because as I said reality does have a mathematical structure to it. To call that "not reality" is just incoherent, the structure of reality is obviously part of reality.
• 520
Reality follows patters which we can infer through scientific observation and measurement.

However the math that we humans have invented to approximate it is not reality.
hks

I think you are confusing our description of reality with reality. It is true that math is made up by humans, or a game we play or however you want to put it but its also true that seperate from that “game”, there are actual numbers of things and more and less of things etc
Math, or perhaps more properly advanced math/physics, becomes this “game” as it creates models and such (that work, thats important) but I don’t think becuase of that you can make the claim you are. There is room for both things to be true.
Also, to say something that is part of reality “isnt reality” seems confusing. Are you making a distinction between reality and something that exists in reality when you use the term “reality”. Seems like you are saying the two are separate rather than one encompassing the other, is that right?
• 79
Restricting Determinism to direct physical objects is a straw man.

Laplace's original formulation concerned physical law and matter and that is what I am addressing. Mathematical truth is not physically determined by any physical state in the universe. I am not talking here about applied mathematics, I am talking about pure number theory. It is what it is, eternally. That is what is important. Eternal mathematical truth determines what happens, not any physical state.

So basically irrationality or what the number would be used here wasn't important. "Known in advance" is quite vague definition here. By whom?ssu

By the experimenter. It is necessary that the digit is not known in advance to counter those who say it is having an unconscious influence.

This is a bit confusing. How do you define these two to being "physical", yet then something being "mathematical" as opposition to the first?ssu

Pure mathematics - as opposed to applied math. - is true, regardless of any physical state.

Those are still determinate events since you could say it was your neurological (clasical) state which chose the irrational number.

Yes, the original starting number, 11, but not the value of its decimal expansion. One could choose the millionth digit of the expansion of the square root of 2 and that would get around any objection re. 'brain states'.

No, because as I said reality does have a mathematical structure to it. To call that "not reality" is just incoherent, the structure of reality is obviously part of reality.

See above re. the difference between applied and pure math. My point is that the value of the digit is not determined by any physical state, yet the digit determines the choice from the list. Therefore the choice is not physically determined. The value of the digit cannot be determined by any physical state - it is what it is beyond matter, space and time. (But as I said, above, my only argument is against physical determinism which argues that every physical state is determined by a previous physical state. But if the choice is determined by a digit, the corresponding event is not physically determined.)
• 520
Laplace's original formulation concerned physical law and matter and that is what I am addressing. Mathematical truth is not physically determined by any physical state in the universe. I am not talking here about applied mathematics, I am talking about pure number theory. It is what it is, eternally. That is what is important. Eternal mathematical truth determines what happens, not any physical state.

Ok, in that case Im sorry to say you have posited nothing here at all. Certainly nothing interesting.
You defined the premiss as being outside the physical state of the universe and conclude that a specific equation based on that premiss is outside the physical state of the universe. You can’t have it both ways, either you bring in the choice part and at least some part is dependent on physical states or you do not. Using the argument quoted, you have chosen the latter. Thats fine, but then you arent really saying much at all here.
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