Foundations of Logic – Muhammad asked “Is metaphysics possible?”
Robert replied – “Yes, of course”
This explains completely why Robert & I can agree about very little.
For me, it seems clear from evidence that metaphysics as it is generally conceived is illusion, and that the very idea of a priori knowledge is an illusion gained from a failure to understand the complexity of the evolutionary process that seems (beyond any reasonable doubt) to be responsible for the existence we experience.
Evolution does not require perfection of anything, it only requires that whatever emerges is more capable of surviving in context than any of the other things around. Given the vast array of contexts, and the very tight limits on time and energy available for computation, the emergence of simplistic approximations to complexity is to be the expected norm. Hence very simple ideas like true and false, right and wrong, tend to dominate, and more complex and subtle ideas that take far longer and require vastly greater computation are rare.
Hence the simple idea of “a priori” knowledge is what one would expect to emerge early in the process of gaining some beginnings of an understanding of the evolution of consciousness and knowledge and the capacity for abstract thought (which is an extremely complex and multi leveled set of understandings and approximations to something that seems to be complex beyond the capacity of any human mind to know in detail – only by rough approximation and analogy).
Various forms of mathematical and logical conjectures are possible and can be derived from sets of axioms, and some of those give very reliable results when applied to some sets of experiments, in some contexts. But, there is no method other than experiment to determine which of the infinite class of possible systems actually gives the best approximation to whatever reality actually is in any particular context.
Nor does it seem to be necessary that reality necessarily follow any set of conjectures in all possible contexts.
It does in fact seem entirely possible (though not provable beyond all doubt), that this reality we find ourselves in is, in some set of fundamental ways, some balance between the lawful and the random – some sort of probabilistically constrained randomness.
[followed by – Robert responded with … “Then provide any context where the foundational Truths of Mathematics such as the Continuum Hypothesis do not apply/ follow exactly.
You obviously will fail to provide any such context or scenario with any such counterexample.”… ]
Such is clearly your unquestionable belief.
[followed by Robert responded “Put up or shut up.”]
I do not, and have never, denied that within the postulates of ZFC, the continuum hypothesis is provable.
I have not done the work to either confirm or deny it.
You seem confident that it is proven.
I am certainly not denying that.
From my perspective, the conversation has never been about the Truth’s of derivations of ZFC or any other set of postulates, but of the idea of truth in respect of reality (whatever reality actually is).
In another sub-thread below you use the term “a priori necessary conditions of the understanding”.
That term seems to have a rather strict meaning for you.
For me, the idea of anything being “a priori necessary” is not required.
We seem to be the result of a recursive process of evolution.
Our ability to understand anything (at least to the degree that we do) seems to derive from the ability of that process to select those systems that survive better.
In terms of the evolution of brains and the ability to model and abstract, there have been strong pressures to use heuristics that are good enough to survive.
Most survival contexts have limits of time and energy available for computation.
Many contexts have very tight constraints of both time and energy, demanding simplification of the complexities obviously present.
Thus we expect evolution to deliver neural networks that are strongly biased to the acceptance of the simplest model that works (recursively – all levels).
In exploring the space of all possible ontologies, it does not pay to get stuck on the first and simplest one that one encounters.
It is not necessary that reality obey the rules of logic for complex systems to evolve, only that it approximate them to some useful degree of reliability in some set of contexts.
If one is dealing with a single “tick” of a cesium clock for example, one is dealing with some 10^30 instances of Planck level existence. If that Planck scale quantum existence is actually random within constraints, then a distribution populated with some 10^30 instances will have a very reliable form.
In such a manner, constrained randomness can deliver a close enough approximation to causality for the sorts of systems that we are to emerge, and for the equations of quantum mechanics to give us useful probabilities about the sorts of things we should expect to see at that scale.
Does that mean that “reality” follows hard rules?
It is possible that everything follows hard rules and that some of the variables behind those hard rules are hidden from us.
It also seems possible that real randomness may be at play in the game.
If it is the latter, then it seems to be a very different sort of game from the one that you seem to be insisting always follows the rules of a set of postulates (rather than simply approximating them in practice).
All I am stating, is that in my experience, my individual investigations, the latter seems to be a likely scenario, and seems to hold greater promise for notions like “freedom” and “responsibility”.
It seems to be a different way of conceptualising the ability to conceptualise.
Clearly, you have not understood the argument I have proposed.
I am not arguing about mathematics as a logical construct, or any truth expressible in it, or any form of logic. I accept that such forms of truth follow from the stated assumptions of those systems.
What is at issue, is the idea of proving that the relevant sets of assumptions necessarily apply in all of “reality”.
That cannot be done.
It is a logical impossibility.
It seems beyond all reasonable doubt that we do not and cannot have access to all of “reality” to be able to test and make such a claim.
We are inside of reality, not outside of it.
Vast evidence sets indicate that we are evolved cooperative entities with brains with neural networks with large sets of biases that aided the survival of those systems in our past, but are not necessarily well adapted to our exponentially changing present and future.
Being able to distinguish that what we experience as reality is a vastly simplified model of reality created by our neural networks and associated sets of molecular and computational and cultural system, is an essential step in moving towards understanding.
We cannot prove the constraints of the system that we are ourselves embedded in.
We can search the space of all possible systems, looking for those systems that provide useful explanatory power when applied to the datasets of observations we have available. And sometimes the mathematical and logical implications of those systems give use useful approaches to designing new experiments to look for things we had not previously noticed.
But in the set of all possible mathematical and logical systems, only a very small subset seem to have any sort of fundamental utility in explaining what we observe. And it isn’t at all obvious just from looking at mathematics and logic why any of those particular systems or numbers work as well as they do in the experimental results we observe.
So my argument is, and always has been, about the applicability of anything to reality.
Reality seems to have many sets of fundamental constraints on the degree to which we can know it, or approximate what it is and how it works at the most detailed levels.
It is in this sense, and this sense only, that I deny the utility of the absolute idea of “Truth”, as distinct from its weaker cousin “truth” (being a contextually useful approximation that works in practice in the contexts we have explored thus far and to the degree that we have explored them – which in my world occurs more usefully as “heuristic”).
The game of go is played on a 19 x 19 matrix and has about 10^172 possible positions – How is the continuum hypothesis relevant to and provable within the space of “go”?
The reality in which we exist is certainly a much larger game space than that of go.
The evidence we have seems to indicate that the reality within which we exist has a finite number of particles within any particular time cone of space-time, even if it is larger than any space-time cone can traverse/encompass.
How can you prove the relevance of CH to that finite entity in all cases?
What evidence do you have that proves that assertion true over all of reality?
What is the 1:1 correspondence present between CH and reality? The truth value of the assertion in respect of reality.
I cannot decide what you mean by the continuum hypothesis.
I take it to mean:
This talks about infinite sets.
I am talking about reality, which seems to be a finite set (though very large, and beyond physical counting – as evidence indicate it is wider than light/information can travel).
I therefore fail to see how one can use CH to describe some aspect of reality as being true.
I fail to see a 1:1 correspondence between the real universe within which we seem to exist, and CH.
If you have one, please say what it is.
And the fact that reality seems to be wider than information can travel prevents us from making any absolute claim about reality that is stronger than probability, as we cannot fully test it.
The subtle, and not so subtle, difference between us seems to be in the use of assumptions.
For you, it seems OK to apply assumptions to reality.
For me, it never is.
I acknowledge all the power of the assumptions of mathematics and logic. They give us very powerful modelling tools.
However, there is no requirement that reality (whatever it is) actually adhere to them in all cases.
Thus, when it comes to reality, it is experiment that trumps deduction – every time. And certainly, deduction can and has led to some great experimental design.
The Pythagoreans thought they had perfect answers in geometry, in the circle.
Newton thought he had the answer – perfect equations, elliptical orbits. Correct to within the limits of experimental error.
QM and relativity work to within the limits of experimental error.
Is that reason to think them perfect?
Not in my world.
Useful approximations – yes – certainly that.
When building a house, I use flat earth (near enough to work).
When piloting aircraft I use Newtonian mechanics (works fine for navigation, within the errors of measurement available).
Building a GPS network requires relativity and QM – relativity for the time adjustments wrt orbits and QM for design of the chips used.
Things can be useful, without having to be perfectly true.
That seems to be the lesson of biology and evolution and history.
Perfect Truth can exist in the realms of mathematics and logic.
The realms of mathematics and logic are great tools for building our understandings of reality.
The evidence we have from reality and history is that our tools tend to give us useful approximations to whatever reality is.
We have no evidence for anything more than that.
We have a great deal of evidence for uncertainty at every level.
We have a lot of evidence for the idea of useful approximations, and a lot of falsification of “Perfect Truths” in respect of reality.