The Relationship Between Math and Physics

Science
Philosophy

(Dan Eastwood) #61

Yes, but if you want to jump in, it goes south long before this.


(Mikkel R.) #62

How’s that for non-sequitur of the year?


(Evan) #63

@AntoineSuarez

Are you referring to the types of arguments given by Augustine and Leibniz about eternal truths leading to the existence of God? I think that may be why there is a disconnect between you and some of the other readers


#64

I am feeling a bit guilty about posting the Borges story without more detail on his background: perhaps some may think I was attempting of Sokal-style hoax at your expense. But such was not my intent at all: I just put it out as food for thought.

Let me give you my detailed take on your argument.

There are at least three issues built into this premise:
First, can we prove using the axioms and rules of any formal system that all theorems stated under that formal system are either true or false? The answer is no: that is what Godel’s incompleteness results prove.

Second, are there ways to nonetheless ascertain the truth of theorems outside of that formal system. The answer is yes: that also is part of his proof.

Third, does that result imply that minds have ways of assessing truth of theorems in formal systems which go beyond the methods available in formal mathematical systems? Or can we only assess the truth by adding more axioms to the system and then applying the usual rules of proof?

Godel and Lucas and Penrose thought/think minds do have ways of assessing truth which go beyond formal proof, but as detailed at the linked IEP entry, that view is mostly rejected by philosophers and mathematicians who have studied the issue.

So I don’t think this premise is nonsense. But I do think it is false and so do most experts who have looked at it.

This is definitely false. Algorithms have to halt by definition and Turing directly (and Godel indirectly) showed that there are problems which have answers but which do not have algorithms (which are part of formal systems by definition) to solve them.

But you do not need that premise to believe that mathematicians have non-algorithmic ways of accessing truth. (By the way, if you have been following the EricMH threads, such an ability might constitute a halting oracle).

I think this is the heart of your argument.

I take answers to be abstract objects. That is, they do not exist in the spatio-temporal realm that everyday things exist in (things like DNA, rocks, and human brains). Do such abstract objects exist in some other realm? I think not. But many philosophers think they do, eg mathematical platonists think mathematical entities exist (Godel was a math platonist). So this part is not nonsense, though many reject its truth (eg via Neil’s fictionalism).

Now does existence of abstract objects imply they exist in some mind? That seems to me to be a form of idealism: as the linked SEP article explains, this is the belief that objects owe their existence to the mind. It follows from idealism that if objects exist but no human mind has yet conceived them, then they must exist in some other mind. God’s mind must exist to provide the ground for all existing objects.

I think that the belief in the existence of abstract objects and the need for a mind to ground the existence of any object is all you need for your argument. I doubt both premises. But I don’t think they are nonsense, just false.

You may enjoy this limerick if you have not yet seen it:


(Antoine Suarez) #65

Thanks Bruce!
This is a serious and constructive comment, which deserves a detailed answer.
I will come back to you in 3-4 days, when an ongoing meeting finishes.
Meanwhile all the best
Antoine


(Antoine Suarez) #66

I am rather referring to Moses Mendelssohn’s proof of the existence of God.
Unfortunately Mendelssohn’s was “euthanized” by Immanuel Kant.
My proof proves that Kant was wrong and Mendelssohn right.
More in 4 days.
Antoine


(Antoine Suarez) #67

I would be honored to receive this award.
But you have first to prove by arguing seriously that I deserve it.


(Dan Eastwood) #68

The Non-Sequitur of the Year Awards Committee always conducts a rigorous review of all candidate submissions, selecting the best of the best. Then they give the award to someone else, seemingly at random.

:smile:


#69

I was not familliar with that, but from a brief look at the description at SEP, I understand that Moses M made an ontological argument that involved taking existence as a predicate, that he was aware of Kant’s rejection of existence as a predicate, but he explained why he thought Kant was wrong.

But, frankly, ontological arguments make my brain hurt.

So it would be helpful to me if you explain your understanding of Kant and Moses M as part of your reply, before detailing why you think your argument supports Moses M.

https://plato.stanford.edu/entries/mendelssohn/#NatThe

ETA: If you are really feeling industrious, perhaps you would describe how your argument deals with the relevant objections from this list:
https://plato.stanford.edu/entries/ontological-arguments/#ObjOntArg


(Antoine Suarez) #70

With pleasure. I hope to find time during the weekend. Busy days. I apologize.


(Antoine Suarez) #71

The Philosopher Moses Mendelssohn published his Lectures on the Existence of God in year 1785, months before he died. A main motivation of Mendelssohn was to promote the Haskalah (Jewish Enlightenment), to show that the Jewish religion is compatible with the ideal of reason. His Lectures were published in Berlin under the German title Morgenstunden (Morning Hours).

Toward the end of Morgenstunden Mendelssohn introduces a proof of the existence of God, which is based upon the basic incompleteness of our self-knowledge, and of human knowledge more in general. He concludes by stating: “So we would have a new scientific proof of the Existence of God that is based upon the imperfectness of our self-knowledge”.

The proof can be summarized as follows:

Premise A.
“There is more to my existence than I might consciously observe of myself”. Even what I know of myself (and my mind) is capable of far greater development, and greater completeness than I am able to give it.

Premise B:
“Everything actual must be thought to be actual by some thinking being”. Reality is thinkable because it is thought.

From Premises A and B Mendelssohn infers that:

Conclusion C:
There is “an infinite intellect” that does represent everything to itself and contains all the knowledge that is not contained in the human minds.
(See Gesammelte Schriften 3/2, p. 141-147)

Regarding the meaning of Premise A:
It seems sound to assume that at any time of history there will always be knowledge about actually existing things, which does not exist in any human mind. Consider for instance the search for extraterrestrial life. Nobody knows whether or not there is life in other planets; this knowledge is not contained in any human mind for the time being. In this sense what I know of my mind at any given moment is capable of greater completeness than I am able to give it.

Regarding the meaning of Premise B:
Consider software for editing videos. You get the expertise for this by reading attentively the instructions. These instructions are knowledge which exists in the mind of the computer scientist before it comes into your mind.

For Mendelsohn all the knowledge we can get, also the knowledge about the world outside there, is like software or information that is contained in a mind even before we get hold of it. Mendelssohn claims: “All what is real must be not only think-able, but must also be thought by some mind” (“alles Wirkliche muss nicht nur denkbar sein; sondern auch von irgend einem Wesen gedacht werden.“). Reality is thinkable because it is thought.

On his turn, the philosopher Immanuel Kant strengthened Mendelssohn’s Premise B on the basis of mathematical knowledge.

According to Kant mathematics is the highest form of knowledge a priori: A mathematical theorem contains nothing more than the ingredients the mathematician puts into it. Mathematics is knowledge about the way our minds function, our self-knowledge regarding forms and modes of mental operations.

Kant’s deep insight can be better understood by considering the operation of counting. It is a mental operation which generates the natural numbers. Numbers are mental data as strongly real as sense data are. So for instance one can distinguish even, odd, prime numbers etc., the same way as one distinguishes different species of animals.

Consider now prime numbers. A prime number is a natural number greater than 1 whose only factors are 1 and itself.

One can then ask: Are there infinitely many prime numbers? The answer exists: it is either YES or NO. Euclid (300 BC) proved that the answer is YES through the following argument:

Let us assume that the set of prime numbers is finite.
Then we can write down the list of all of them:
p1, p2, p3, …, pn.
Then we construct the number p by multiplying all the prime numbers in the list and adding 1:
p = p1 x p2…x pn +1

The number p is different from any number in the list, and therefore by assumption it cannot be a prime number. Hence there must be a prime number in the list that divides p.

However, no prime in the list divides p, otherwise it would also divide the difference
p - (p1 x p2…x pn) = 1
and divide 1, which is impossible.

So there must be another prime number which is not in the list and divides p. This is in contradiction with the assumption that the list contains all prime numbers.

So the only way to escape the contradiction is to accept that our assumption at the beginning is false, and conclude that there are infinitely many prime numbers.

Euclid’s proof illustrates quite well the nature of mathematical knowledge. The proof consists in a number of mental operations showing that a certain assumption leads to a contradiction. Mental creativity and the principle of non-contradiction lead to mathematical evidence, in a similar way as experimental creativity and observation lead to physical evidence.
Mathematics is knowledge about the modes and forms of mental operations with numbers.

On the other hand, Kant states that knowledge a priori is what gives thrust to our knowledge. In support of this he refers to Galilei’s experiments “with balls of a definite weight on the inclined plane”. Indeed, the physics of Galilei and Newton was showing that we can describe the world by means of numbers and equations and thereby figure out what is going on.

Mendelssohn declared: “Each evidence based on sense data, corresponds necessarily to a truth of reason”. Kant’s knowledge a priori strengthens Mendelssohn’s insight: What is observable is thinkable. This was the beginning of the view that the world is information after all, which is the prevailing view in today’s quantum physics.

And so it was natural to ask, as Mendelsohn and Kant did, where does this information come from. If the world is essentially thinkable this means that it is a mode of thinking of some mind, as Kant stated. So both Mendelssohn and Kant concluded: reality is intelligible or thinkable because it itself is thought, emerges from a knowing intellect.

But while Mendelssohn concluded that reality must be actually thought by God, Kant, in his Critique of pure reason, rather suggested that the world is thinkable because it is thought by me. So Kant proposed to perform a Copernican revolution in Metaphysics and postulate that my knowledge does not follow from reality but reality follows from my knowledge.

In summary both Mendelssohn and Kant agree in that mathematical knowledge is knowledge about our mind and in this sense self-knowledge. According to both the answers to mathematical questions are mental contents and as such are knowledge a priori, i.e.: they exist in a mind.

However, whereas Mendelssohn claims that our mathematical knowledge will always be incomplete, Kant somewhat suggests that mathematics is completely contained in our mind, and thereby rules out Mendelssohn’s Premise A, and consequently the Conclusion C.

For Mendelssohn our mathematical knowledge will always be limited, and therefore the answers to mathematical questions exist in God’s mind.

For Kant mathematical knowledge is contained a priori in the human mind. Mathematical theorems don’t contain more than what our minds put into them.

Now, the Undecidability theorems prove the intrinsic incompleteness of the mathematical knowledge human minds will ever reach. These theorems have been established mainly by Kurt Gödel (1931) and Alan Turing (1936-37).

Accordingly, mathematics is certainly knowledge a priori but in God’s mind, not in our minds. I think Mendelssohn’s proof can stimulate very much the current discussion on God’s existence.

A final remark:
The preceding proof is absolutely NOT an ontological argument based on logical properties of some concept we construct. It is an argument based on the real existence of human minds and the real limits of human reasoning.

Actually it is the argument the advocates of ontological proofs were longing for: A proof on the basis of the reality of thinking that completes the cosmological proof on the basis of the reality outside there. The ontological proofs went wrong because you cannot derive existence from logic. By contrast you can derive logically existence from existence.

So in the light of Turing’s Undecidability theorem (no “solution” for the Halting problem) Mendelssohn’s proof can be considered computational metaphysics.

I would like to finish wishing
Merry Cristmas to those who believe that the Word became flesh and thereby defined humankind as being in God’s Image,
and
Happy New Year to all of you.


#72

I don’t agree with your characterizations of Kant, Godel, or Mendelssohn and ontological arguments (at least as the latter two ares portrayed by SEP). But those disagreements don’t matter for my feedback on your argument as stated in this post, so I will not detail them.

I understand this as saying there are facts regarding my nature which I can never consciously observe and hence can never know. I am not sure if that is true, although it may be true with regard to introspection (without being true with regard to inter-subjectively produced scientific knowledge). But in any event, I do not see this premise as needed in your argument.

I read this as claiming two related things:

  1. Every entity in the actual world exists because of the thought of some being that it is actual.
    I don’t accept this premise. As I said previously, it seems to be a form of idealism, which I reject. I say the existence of things does not depend on the existence of minds, let alone their thoughts.

  2. Reality is thinkable because it is thought.
    I’m not sure what to make of this. First, it seems to be confusing whether it is possible to think about reality with reality itself. Second, “thinkable because it is thought” strikes me as at best a unhelpful truism, eg thinking produce thoughts.

Your conclusion seems to require only the premise that for something to exist, some being must have knowledge of it. Perhaps that premise is contained in your premises, but regardless I think stating it my way captures the essence of your argument. I disagree with such a premise. Things can exist regardless of the existence of minds and their thoughts.

Some of your examples refer to mathematical entities. These bring in the unstated premise that math platonism is true, that is that math objects exist. I also reject this premise.


(Antoine Suarez) #73

This is an excellent suggestion!

Effectively, our knowledge unfolds through questions, and can be considered to consist in answers to questions.

And this is precisely why reality is thinkable: Reality is made so that it fits to our questions.

As you very well state “the question itself would not exist without a mind”.

Accordingly, reality presupposes some mind who understands all possible questions we can ask and has answers to them.

A limited mind (as the mind of any human) contains unanswered answerable questions .

An omniscient mind (God’s mind) contains actually the answer to every answerable question .


(Dr. Patrick Trischitta) #74

There isn’t any omniscient mind. It is a figment of your imagination.


(Neil Rickert) #75

It seems more likely, that we make up our questions to fit reality. But, of course, not always – some people do ask questions that don’t fit.


(Bill Cole) #76

If you can make the case of how conscious intelligence arose without a conscious intelligence being the cause.


(John Harshman) #77

Sure, if you first make the case of how conscious intelligence arose with a conscious intelligence being the cause, and of course how that conscious intelligence arose itself, and so on.


#78

Two objections:

  1. There are parts of reality that do not “fit” to any questions we can ask. For example, the parts of reality that lie beyond the observable edge of the universe). We may invent questions about what might lie beyond that edge, but we will never observe any information about what that reality actually is.
  2. It is unclear to me why reality has to be constrained by human imagination. There are a myriad of conjectures about unexplained aspects of reality, most of which will prove to be wrong when we get around to collecting evidence to settle the questions. I can ask questions about unicorns or gods, but simply because such things are “thinkable” does not conjure them into reality.

(S. Joshua Swamidass) #79

Can you make the case that conscious intelligence arose by conscious intelligence? Hehe. Seems like no one really knows what consciousness is, much less how it arises.


(Bill Cole) #80

Agree. That was Nagel’s point in mind and cosmos. To have 7 billion human conscious minds in the world and to assume something less capable caused their origin seems like faulty reasoning. Especially with all the empirical evidence against this hypothesis.