Incompleteness, Undecidability, and Uncertainty

[0r0r]

It'll take me some time to consume what you've written there but, put as simply as I can, 'existence' and 'reality' are our human (English) words. We control their meanings. The way we go about assigning meanings, is all important.

There are two ways; either by belief or, via the scientific method. The word may be the same, but the meanings are vastly different because of the way we go about choosing meaning.

No matter which way one looks at it though, the meanings of 'existence' and 'reality' have the human mind's fingerprints all over them .. There is no reasoned way know of (yet) for concluding that these meanings came about independently from human minds .. other than by just believing that. The idea that things exist independently from any/all human minds is just another belief.
There is no way known yet, that we can escape from the 'prison' that everything we perceive, leaves behind objective evidence of some active human mind at play and none for the idea that 'things exist' independently from some human mind or other.

The outlined thinking above is entirely consistent with the scientific method and scientific thinking. It is in fact, demonstrably, the philosophy of Science, in a nutshell.

I'm not trying to squeeze out the mind. In fact, one of the central theses of Gödelian Incompleteness, Platonism, & the Mind of God is that the mathematical structure of the world depends on God's mind.

 

[SelfSim]

I'm not trying to squeeze out the mind. In fact, one of the central theses of Gödelian Incompleteness, Platonism, & the Mind of God is that the mathematical structure of the world depends on God's mind.

I don't understand what you mean by 'God's mind' .. but I can be sure that it'll depend on your mind.

 

[FrumiousBandersnatch]

The crux of the matter is as follows: If axiomatic provability and mathematical truth are not the same, then in what does mathematical truth inhere? It is natural to think that a mathematical statement is true because it is provable, but Gödel proved that the set of provable mathematical statements is a proper subset of the set of true mathematical statements. Therefore, mathematical truth cannot be tamed by axiomatics.

The unprovability or undecidability of a specific subset of mathematical statements doesn't render all mathematical statements unprovable or undecidable any more than logical paradoxes render logic unusable. It applies to formal systems in general - I don't see how that suggests they all have some kind of objective reality.

 

[FrumiousBandersnatch]

I don't understand what you mean by 'God's mind' .. but I can be sure that it'll depend on your mind.

If you ask Spinoza, he'd say it's the laws of physics...

 

[0r0r]

I don't understand what you mean by 'God's mind' .. but I can be sure that it'll depend on your mind.

That would only follow if mathematical truths depend on the human mind, which is exactly what Gödel disproved.

My argument, in outline, is: The eternal (or immutable) truths of mathematics could only root in the Necessary Being. If you wonder what the argument for the Necessary Being is, see my essay on Gödel's proof of the existence of God. I also offer a monotheism theorem, which excludes the possibility of more than one Necessary Being.

 

[SelfSim]

That would only follow if mathematical truths depend on the human mind, which is exactly what Gödel disproved.

When Gödel was working on his Incompleteness Theorem, he was using his mind. His theorem therefore was not independent from his mind.
The same goes for all mathematical axioms .. they were all conceived and described by human minds in the syntax and tautolgy of math .. all human constructs. There is abundant historically documented (objective) evidence for this.

The leap you are making, which is untestably speculative, is that the incompleteness proof means that there exist mind independent truths. No-one, thus far, can even produce a truly mind independent objective test for that. Your conclusion is thus (objectively) flawed. You can believe the human mind independence of them .. but this is not objective evidence of that until you can truly discount the influences of human perception in some test or another.
I am thus objectively justified in claiming that the notion is just another belief and is thus, very much mind dependent.

My argument, in outline, is: The eternal (or immutable) truths of mathematics could only root in the Necessary Being. If you wonder what the argument for the Necessary Being is, see my essay on Gödel's proof of the existence of God. I also offer a monotheism theorem, which excludes the possibility of more than one Necessary Being.

And I shall notice all the way through reading it, the evidence you have left behind that your mind was at work in producing it. You may think this as being trivially obvious .. however, the accumulated objective evidence will be overwhelmingly in support of the mind dependence of your conclusions on one hand .. and zero evidence supporting the notion of mind independence, on the other.

The so-called 'immutable truths of mathematics' you mention, have also left behind overwhelming objective historical evidence of the minds who devised them. To me, the axioms pretty much stand as a testament to fundamental notions that any healthy logically thinking human mind would agree on. Which is pretty amazing .. given how much trouble we normally have on agreeing on just about anything!

 

[0r0r]

When Gödel was working on his Incompleteness Theorem, he was using his mind. His theorem therefore was not independent from his mind.
The same goes for all mathematical axioms .. they were all conceived and described by human minds in the syntax and tautolgy of math .. all human constructs. There is abundant historically documented (objective) evidence for this.

The leap you are making, which is untestably speculative, is that the incompleteness proof means that there exist mind independent truths. No-one, thus far, can even produce a truly mind independent objective test for that. Your conclusion is thus (objectively) flawed. You can believe the human mind independence of them .. but this is not objective evidence of that until you can truly discount the influences of human perception in some test or another.
I am thus objectively justified in claiming that the notion is just another belief and is thus, very much mind dependent.

And I shall notice all the way through reading it, the evidence you have left behind that your mind was at work in producing it. You may think this as being trivially obvious .. however, the accumulated objective evidence will be overwhelmingly in support of the mind dependence of your conclusions on one hand .. and zero evidence supporting the notion of mind independence, on the other.

The so-called 'immutable truths of mathematics' you mention, have also left behind overwhelming objective historical evidence of the minds who devised them. To me, the axioms pretty much stand as a testament to fundamental notions that any healthy logically thinking human mind would agree on. Which is pretty amazing .. given how much trouble we normally have on agreeing on just about anything!

I think we should talk in greater detail about the Incompleteness Theorems, but not now. First, let me address the issue that keeps popping up, namely: the entanglement of mind and reality.

I'm not necessarily opposed to Bishop Berkeley's thesis esse est percipi. Of course, considerations of continuity led to the now famous limerick:

There was a young man who said God,
must think it exceedingly odd
if he finds that the tree
continues to be
when no one's about in the Quad.

Dear Sir, your astonishment's odd
I am always about in the Quad
And that's why the tree
continues to be
since observed by, Yours faithfully, God.

I would, however, go one step further than Berkeley in suggesting that the mind of God is the abode of not only all actuals but indeed all possibles, where an object is possible if it does not entail a logical contradiction. If pressed, I would say that the domain of possibility has a reality of its own, though it is not, strictly speaking, part of actuality. If pressed further, I would have to say I am not prepared to elaborate just yet. This distinction between possible and actual must, in some way, pivot on the divine will, as opposed to the divine intellect. More than that, I cannot say at the moment.

It is pleasing to note that our conversation thus far has provided ample evidence of my original thesis that the rational enterprise is inextricably linked to traditional philosophy.

 

[0r0r]

I think we should talk in greater detail about the Incompleteness Theorems, but not now. First, let me address the issue that keeps popping up, namely: the entanglement of mind and reality.

I'm not necessarily opposed to Bishop Berkeley's thesis esse est percipi. Of course, considerations of continuity led to the now famous limerick:

There was a young man who said God,
must think it exceedingly odd
if he finds that the tree
continues to be
when no one's about in the Quad.

Dear Sir, your astonishment's odd
I am always about in the Quad
And that's why the tree
continues to be
since observed by, Yours faithfully, God.

I would, however, go one step further than Berkeley in suggesting that the mind of God is the abode of not only all actuals but indeed all possibles, where an object is possible if it does not entail a logical contradiction. If pressed, I would say that the domain of possibility has a reality of its own, though it is not, strictly speaking, part of actuality. If pressed further, I would have to say I am not prepared to elaborate just yet. The distinction between possible and actual pivots on the divine will in some way, not just the divine intellect.

It is pleasing to note that our conversation thus far has provided ample evidence of my original thesis that the rational enterprise is inextricably linked to traditional philosophy.

 

[0r0r]

The unprovability or undecidability of a specific subset of mathematical statements doesn't render all mathematical statements unprovable or undecidable any more than logical paradoxes render logic unusable. It applies to formal systems in general - I don't see how that suggests they all have some kind of objective reality.

To be clear, nothing in my work suggests that Gödel's Incompleteness Theorems imply the unprovability or undecidability of all mathematical statements. That is not an implication of Gödel's work, and it is not a view that I have ever held.

For every consistent system S that entails a certain amount of arithmetic, there are true statements of S not provable in S. Since axiomatics cannot exhaust mathematical truth, it stands to reason that mathematical truth must root in something besides axiomatics.

I want to show clearly how Gödel's work implies mathematical Platonism. To that end, I'm working on an article which, I think, makes a clear case. I'll notify you when its done.

 

[SelfSim]

I
I'm not necessarily opposed to Bishop Berkeley's thesis esse est percipi. Of course, considerations of continuity led to the now famous limerick:

There was a young man who said God,
must think it exceedingly odd
if he finds that the tree
continues to be
when no one's about in the Quad.

Dear Sir, your astonishment's odd
I am always about in the Quad
And that's why the tree
continues to be
since observed by, Yours faithfully, God.

So, along similar lines there (I think?) and more where I'm coming from, is the old conundrum:
- 'If a tree falls in the woods, does it make sound'?
The answer is of course it does, because what we mean by a tree, doing what we mean by falling in a woods, does what we mean by making sound. The human mind dependence of all of that could not possibly be more obvious, it's a hypothetical tree for crying out aloud!

I would, however, go one step further than Berkeley in suggesting that the mind of God is the abode of not only all actuals but indeed all possibles, where an object is possible if it does not entail a logical contradiction. If pressed, I would say that the domain of possibility has a reality of its own, though it is not, strictly speaking, part of actuality. If pressed further, I would have to say I am not prepared to elaborate just yet. This distinction between possible and actual must, in some way, pivot on the divine will, as opposed to the divine intellect. More than that, I cannot say at the moment.

Meh .. I'm not too concerned by what Berkeley might have to say.
'Logical contradiction' and 'domain of possibility' are what human minds conceive - there is abundant objective evidence for this and none for their independence from that conceiving human mind.
Both 'possible' and 'actual' are what we decide they mean and we have two different ways of going about that;
- belief (justified, 'true', or whatever) or;
- objective science.
In the case you give above, you simply haven't made this demonstrable facts explicit ..
Please understand, thus isn't peculiar to just yourself .. it beats me why we all seem to have some kind of blind spot for the trivially obvious origins of meaning acquisition in language. However this notion goes way deeper than just semantics because we're really talking about how our minds articulate our perceptions and concepts of what we mean by 'nature', 'the world', 'the universe', etc. They are demonstrably all mind models.

It is pleasing to note that our conversation thus far, has provided ample evidence of my original thesis that the rational enterprise is inextricably linked to traditional philosophy.

Or more importantly, we cannot escape the roles, functions, influences and evidence left behind (ie: the mind's fingerprints) by our own human minds when we speak about rational enterprise and philosophies or any other perceptions/concepts.
And there is no such objective evidence, whatsoever, for mind independence there .. so we just seem to go on believing there is ... (even though there isn't any).

Science is a way of thinking too. However, we have to do our very best to track our mind's influence whilst directing the scientific method at questions about existence or reality. Its the best we've come up with for doing this .. its not great, but its still more internally consistent than the other way we do it (IMO).

 

[0r0r]

So, along similar lines there (I think?) and more where I'm coming from, is the old conundrum:
- 'If a tree falls in the woods, does it make sound'?
The answer is of course it does, because what we mean by a tree, doing what we mean by falling in a woods, does what we mean by making sound. The human mind dependence of all of that could not possibly be more obvious, it's a hypothetical tree for crying out aloud!

Meh .. I'm not too concerned by what Berkeley might have to say.
'Logical contradiction' and 'domain of possibility' are what human minds conceive - there is abundant objective evidence for this and none for their independence from that conceiving human mind.
Both 'possible' and 'actual' are what we decide they mean and we have two different ways of going about that;
- belief (justified, 'true', or whatever) or;
- objective science.
In the case you give above, you simply haven't made this demonstrable facts explicit ..
Please understand, thus isn't peculiar to just yourself .. it beats me why we all seem to have some kind of blind spot for the trivially obvious origins of meaning acquisition in language. However this notion goes way deeper than just semantics because we're really talking about how our minds articulate our perceptions and concepts of what we mean by 'nature', 'the world', 'the universe', etc. They are demonstrably all mind models.
Or more importantly, we cannot escape the roles, functions, influences and evidence left behind (ie: the mind's fingerprints) by our own human minds when we speak about rational enterprise and philosophies or any other perceptions/concepts.
And there is no such objective evidence, whatsoever, for mind independence there .. so we just seem to go on believing there is ... (even though there isn't any).

Science is a way of thinking too. However, we have to do our very best to track our mind's influence whilst directing the scientific method at questions about existence or reality. Its the best we've come up with for doing this .. its not great, but its still more internally consistent than the other way we do it (IMO).

I have offered what I regard as incontrovertible evidence of Mathematical Platonism, but I acknowledge that I may not have made the case as clearly as I could have. To that end, I am working on an article that shows quite clearly how (a) Gödel's Incompleteness Theorems imply Platonism and (b) your mind-based objection is no obstacle at all.

 

[SelfSim]

I have offered what I regard as incontrovertible evidence of Mathematical Platonism, but I acknowledge that I may not have made the case as clearly as I could have. To that end, I am working on an article that shows quite clearly how (a) Gödel's Incompleteness Theorems imply Platonism and (b) your mind-based objection is no obstacle at all.

All good with me .. Its been a rather pleasant conversation .. thank you for that and thanks for taking the time to write down how you're thinking about all this.

Oh .. and by the way; its not actually my mind-based objection (I wish it was).
Its actually the scientific method applied to the question of how we figure out what's real.
(Ie: its an hypothesis under test).
Cheers

 

[FrumiousBandersnatch]

For every consistent system S that entails a certain amount of arithmetic, there are true statements of S not provable in S. Since axiomatics cannot exhaust mathematical truth, it stands to reason that mathematical truth must root in something besides axiomatic.

Mathematical truth derives from the axioms used - by definition. That some truths are not provable doesn't change that.

I want to show clearly how Gödel's work implies mathematical Platonism. To that end, I'm working on an article which, I think, makes a clear case. I'll notify you when its done.

OK.

 

[0r0r]

All good with me .. Its been a rather pleasant conversation .. thank you for that and thanks for taking the time to write down how you're thinking about all this.

Oh .. and by the way; its not actually my mind-based objection (I wish it was).
Its actually the scientific method applied to the question of how we figure out what's real.
(Ie: its an hypothesis under test).
Cheers

Many, many thanks for the kind words. I too enjoy our conversations. I certainly hope you might become part of the nascent community I recently launched. Now, back to the issue at hand.

I am still working on an essay wherein I hope to demonstrate a necessary connection between certain metamathematical results on the one hand and mathematical Platonism on the other. As I have many obligations which I cannot ignore, this may take some time. Still, since our conversation often circles back to the relation between mind and reality, I thought it might be profitable to make some preliminary comments concerning that relation.

It seems wrong to distinguish experiential phenomena (appearances) from an external, inaccessible, and extra-experiential reality. After all, there is only one reality, and that reality necessarily encompasses private experience. What is accordingly at issue is something else entirely, namely: a distinction between that which our experience reveals to be coherent and true and that which our experience proves incoherent and false; and this brings us back to a central pillar of medieval Catholic philosophy: adaequatio ad rem. As Saint Thomas Aquinas put it, truth is the conformity of mind to reality. This medieval perspective frames reality as something perfectly accessible to human inquiry, though it readily acknowledges that human inquiry may often lapse into error. After all, though the neurotic are unable to see it, there is nothing to prevent appearances from standing coordinated with facts (under the aegis of reason) in many respect. Similarly, reason demands further recognition: appearances can be deceiving.

The medieval approach, therefore, establishes a via media whereby emphasis is laid on an epistemological criterion of coherence rather than an ontological hypothesis of a mind-independent (and therefore largely inaccessible) domain.

Much else could -- and should -- be said, and much of that would lead us into a coherence theory of truth, which I do prefer over a strict correspondence theory.

 

[0r0r]

Many, many thanks for the kind words. I too enjoy our conversations. I certainly hope you might become part of the nascent community I recently launched. Now, back to the issue at hand.

I am still working on an essay wherein I hope to demonstrate a necessary connection between certain metamathematical results on the one hand and mathematical Platonism on the other. As I have many obligations which I cannot ignore, this may take some time. Still, since our conversation often circles back to the relation between mind and reality, I thought it might be profitable to make some preliminary comments concerning that relation.

It seems wrong to distinguish experiential phenomena (appearances) from an external, inaccessible, and extra-experiential reality. After all, there is only one reality, and that reality necessarily encompasses private experience. What is accordingly at issue is something else entirely, namely: a distinction between that which our experience reveals to be coherent and true and that which our experience proves incoherent and false; and this brings us back to a central pillar of medieval Catholic philosophy: adaequatio ad rem. As Saint Thomas Aquinas put it, truth is the conformity of mind to reality. This medieval perspective frames reality as something perfectly accessible to human inquiry, though it readily acknowledges that human inquiry may often lapse into error. After all, though the neurotic are unable to see it, there is nothing to prevent appearances from standing coordinated with facts (under the aegis of reason) in many respect. Similarly, reason demands further recognition: appearances can be deceiving.

The medieval approach, therefore, establishes a via media whereby emphasis is laid on an epistemological criterion of coherence rather than an ontological hypothesis of a mind-independent (and therefore largely inaccessible) domain.

Much else could -- and should -- be said, and much of that would lead us into a coherence theory of truth, which I do prefer over a strict correspondence theory.