Without it, we would not have brains capable of making mathematical abstractions.
That is not as I see it.
It seems that there is an infinite set of possible logics, and an infinite set of possible theorems.
We are exploring low integer instances of most of those domains.
They are conceptual constructs.
Some of them have predictive utility in modeling aspects of this reality we seem to find ourselves in.
Our ability to conceptualise is the result of evolution.
The things that we are conceptualising are evolving.
Some of us are instantiating ever more abstract systems, and that is itself an evolutionary process.
It seems that everything is an evolutionary instantiation of systems, of relationships of information.
Evolution does seem to have a very real claim to being the foundation of mathematics – at many different levels.
Mathematics is a set of constructs.
We postulate a set of relationships, and then see what results from the application of those rules.
In the first instance it seems that we got the idea of mathematics from counting things in reality.
We then started to wonder about the idea of counting, and the systems that result.
I said nothing about arbitraryness.
Set theory is a set of relationships, based on a simple set of postulates, and certain outcomes happen if one follows that set of rules.
It isn’t “Real” in the sense of physical reality, and it is real in a systems sense.
Reality appears to be something.
Mathematics is the best modeling tool that we have.
Thus mathematics gives us the best models of reality that we have.
That does not mean that reality necessarily follows the rules of any particular mathematics in a particular situation, and it gives us very useful approximations in most situations.
And anywhere that irrational numbers like Pi or e appear in an equation, then all we can ever have is an approximation, as those numbers may never be computed completely – only ever to some useful approximation.
I am not sure why you are being so defensive and aggressive.
I have been using mathematics daily for well over 50 years. I rely on it in many ways. And I am very conscious that in any non-trivial instance it only ever gives us approximations to reality, and that those can be very useful indeed in many situations, and not so useful in some.
In terms of the more purely abstract pursuit, of postulating particular rule sets, and seeing what happens when one instantiates instances of them, and the higher order relationships that emerge; that is something that Wolfram explored nicely in NKS.
Thank you Manju.
I was not aware of that particular piece of background.
My apologies to Robert Mosimann if I misinterpreted his intentions.
What I am trying to point to is not trivial.
It is something that someone who has been thinking about these things for over 50 years considers “interesting”.
I am not a specialist logician or mathematician.
I am by choice a generalist. I have done a lot of things in practice, and explored a lot of theoretical frameworks, so I am very used to taking constructs from one domain to another and seeing how they work in both theory and practice. I have worked as a lab technician, a teacher, a fisherman, a programmer, a consultant, an engineer, a carpenter, an auditor, a politician, a law enforcement officer, a director, and have started and run two moderately successful businesses. Not a master of anything, and with a certain knowledge of what works in practice to go alongside my understandings of theory.
On the theoretical side my undergrad work was in biology, biochemistry and marine ecology major interests, but also some courses in psychology, electronics, physics, chemistry, geology, computing, cybernetics, philosophy. I have studied a bit on my own time, working through Riemann, Hilbert, Einstein, as one line of questioning. Also worked through Godel, and quite a bit of the AI stuff. Looked at some of Wolfram’s work. Lots of other stuff (thousands of hours of it).
So definitely not on the cutting edge of math, and not entirely ignorant either.
And I am reasonably confident about the broad brush picture of the evolution of intelligence, and the subtle ways in which evolution both requires and tunes error rates in duplication systems to provide the level of fidelity that is optimal for survival in the specific circumstances of the history of any particular line of organisms, or substructures within more complex systems.
I am very clear how systems based in constrained uncertainty can deliver a very close approximation to hard causality in large aggregates.
So it very much depends upon what one means when one uses the term “REAL”.
Is one referring to the cosmological conditions that we seem to find ourselves instantiated within?
Or is one referring to some set of logical constructs – some set of rules based in some set of assumptions, and simply exploring the relationships that emerge from that rule set?
And I like the example Wolfram uses, of the simple case of a one dimensional array of bistate elements, where the rules about what happens next can only derive from the state of a cell and its 2 neighbours, and what emerges from rule 30 (of the 256 possible rule sets) is very interesting.
That is a very simple demonstration that things can be lawful yet not predictable, from a very simple system.
How much worse when things are fundamentally uncertain, and only approximate lawfulness in aggregate (which is what QM seems to be telling us about the reality we find ourselves in).
So I am definitely a yes to studying and using mathematics, and to maths delivering the best tools we have to model “Reality” (whatever it actually is), and at the same time I am a strong caution against any hubris that any particular mathematical construct is necessarily useful in dealing with reality in any specific context. Reality seems quite capable of delivering exceptions to any and all models.
And at the same time, I am very conscious of the degree of reliability involved in the computer systems we use to record, transmit and display this and other information (and the fact that quantum tunneling is fundamental to the operation of all of these machines). So it is a very strange mixed world we find ourselves it.
And I am just trying to point to something fundamental about the nature of us, and our ability to explore the sets of postulates, rules, and theorems that are possible (as Turing and others postulated, and Wolfram and others have instantiated).
Hi Robert Mosimann
Then we are operating from completely different paradigms, in a very real sense.
I can certainly see some truth in what you say, from a certain perspective.
Yes certainly, if one accepts a set of axioms, and one is capable of applying the relations of those axioms, then it matters not if one is human or anything else, the relationships implied will be instantiated by the application of the appropriate relationships to specific instances.
In that sense, yes – I entirely agree with you.
However, there is another sense that is implied by the use of the term “REAL”, and that relates to the matrix of our existence.
The idea that the matrix of reality necessarily follows any hard set of entirely predictive rules in all situations seems to be implied in what you have written.
If I am mistaken in that – then please say so.
If I am not mistaken in that, then it seems to me, the foundational assumptions of that perspective have clearly been falsified (beyond all reasonable doubt – by the experimental evidence available).
Thus it really does seem to be clearly the case that it is the process of evolution that has instantiated the conditions that allow for the existence of systems capable of postulating the theorems of set theory (or any other branch of logic or mathematics – be they instantiated in Zermelo or Fraenkel or Godel or anyone else).
And I get it is something of a twist I am getting at here.
What is more fundamental, the possibilities of any instantiated set of axioms (recursively applied and instantiated), or the instantiation of something capable of conceiving of those axioms in practice?
Which of those systems is more fundamental?
Kurt Godel could not resolve CH.
I am not in his mathematical league. It took me a year of study after work to go through his incompleteness theorems to the degree that I was satisfied that there were no errors.
I have not worked on the CH conjecture, nor do I intend to any time soon.
That is not what I am trying to point to.
I am pointing at the uncertainty principles of QM, and what they seem to be telling us about the nature of this reality within which we find ourselves (and the supporting evidence sets of course).
The notion that one can build confidence without accepting certainty seems to be recursively applicable to all domains of conjecture.
Godel’s incompleteness conjectures seem to point in that general direction (though with a distinctly different flavour).
The process of evolution, the differential survival of variant systems in variant contexts instantiated with varying probabilities over time, seems to be what has instantiated us and our abilities to investigate logical conjectures.
And I can get that it can be really difficult to see outside of a box when you are so deeply embedded within it. (And I get that is a large part of why I am not prepared to put serious effort in the CH conjecture at this time.)
You made 3 claims:
“1 this whole discussion followed from your view that mathematics is mere human construct or conception or procedures with no underlying Reality as subject matter”
That is not the meaning I was trying to convey.
I get it is the meaning you interpreted.
There must be a way that you constructed that meaning from what I wrote, but I cannot see what it is, too many uncertainties in all candidate explanations I have explored.
“2 you stated 1 hour ago explicitly that the foundation of mathematics being universally applicable has been empirically falsified”
What I tried to be explicitly clear about is the distinction between mathematics and reality.
The conjecture that reality is knowable with absolute precision seems to have been falsified. It seems to contain uncertainties of several different types and at several different levels. The evidence sets for that are vast.
As I stated earlier, it all depends on what you mean when you say “Real” and you have not made any effort to clarify what you mean when you use the term “Real” – I offered two different possible meanings, and asserted that I could accept one, but not the other.
“3” I haven’t been called out for making false statements.
I stand by my statements.
I do assert that my statements have not been understood.
That doesn’t make them false.
I am also quite clear that I do not understand the relevance of your statement in respect of the issue of cardinality between N & R.
I cannot find the link to understanding reality in an asserted difference between two infinities. You will need to be somewhat more explicit in your asserting if you want me to get what you are pointing at. Right now I don’t understand what appears to be significant in the assertion.
I am not slow, and I do not have your experience in the realms of mathematics and logic. I am not without experience, and mine is substantially less than yours. I have quite explicitly stated that I am a generalist. That choice has strengths and weaknesses (as all choices do). I attempt to be as conscious as possible of both.
I answered the lead question with an assertion from a domain space you appear to be less familiar with. I acknowledge that it probably appears nonsensical to you. Be assured it does not so appear to me.
What I am trying to point to is not simple.
Several very distinct domains (at least 3) seem to be collapsed in most people’s thinking, and I was attempting to make them distinct.
Looked at purely from a systems perspective – as a set of relationships, then mathematics has a set of rules that any entity capable of modeling abstract entities should be able to discover and follow. In this abstract sense, the rule sets can be thought of as existent in a space of potentially discoverable relationships that any entity that looks may find. In that sense, they are not the ontological creation of any particular entity, and they do need to be instantiated in some specific entity to be “known” – that is the definition of knowing.
The sets of rules available to construct systems seem to be infinite.
Those that have currently been instantiate as knowledge in any human mind seem to be relatively small numbers in that infinity.
This may seem unrelated but it isn’t – One of the interesting things to come out of database theory this century is that the fastest possible search for a fully loaded processor is a fully random search. How one most efficiently approximates randomness is a quite different question.
Using fully random searches across multiple domain spaces individuals or systems can find solutions to problems without necessarily “filling in the gaps”. This seems to be what evolution does and it also seems to be what intuition does. We seem to have many levels of such systems instantiated within us in the 20 some levels of cooperating systems embodied in each of us (genetic, cultural and conceptual).
What I am clearly saying, is that one can make the strong case from a systems perspective that evolution is the most fundamental construct in the construction of mathematics and logic, for without it there could not exist systems of sufficient complexity to enquire into the nature of relationship itself, and explore the sets of conjectures present in the realms of logic and mathematics.
It is an alternative framing of the question that incorporates cosmology and biology from a systems perspective.
I simply offered it as an alternative.
I wondered if anyone else would get it.
It seems not.
See my reply to Farhan above for part of the answer.
For the other part, I have studied the history of philosophy and mathematics and understanding, and have seen a succession of people make claims as to the the world following simple mathematical rules, only to later find the rules to be approximations to something more complex. It would not surprise me if that process continued indefinitely – that does seem to be what some interpretations of QM point to.
I accept that mathematics and logic have the forms and rules that they have.
I accept that both are potentially infinite.
I accept that they give us the best modeling and predictive tools we have (in those instances where prediction is possible). I also accept the notion of maximal computational complexity, such that even in a fully deterministic world, some things cannot be known by any method shorter than letting them do what they do.
And it seems very probable to me that this universe within which we find ourselves is a very interesting balance between the lawful and the random.
Just as the process of evolution requires a delicate balance of fidelity in copying (too much fidelity and not much happens, too little fidelity and chaos results – there is a context sensitive range in the middle that allows life like us to appear), so it seems that our universe requires some randomness at its foundations, but not too much.
So I think we do have a very fundamental conflict of understanding as to the nature of our being, and the nature of its relationship to mathematics, and perhaps not.
What do you mean by universal?
Within the rules of mathematics, yes: 2+2=4.
In reality – not necessarily.
2 women + 2 men will most likely = 4 people, and there will be distinct and decreasing probabilities of it equaling 5, 6, 7, 8, 9, 10, 11, or 12 people. Numbers beyond 12 while possible are unlikely to be instantiated under normal conditions. Conceptions happen, gametes fuse to form zygotes and new instances instantiate themselves. These events can happen in the time between one reading or writing the left side of the equation and the right side of the equation. Time isn’t universal in that sense, it is both real and local.
Reality has that unsettling ability.
It happens in the sub atomic realm as well.
Something from nothing.
The word “mere” is your own – you introduced it – not me.
It adds a tone I neither intended nor implied.
Please, read very carefully what Ted’s view is:
If we are talking purely about the realm(s) of mathematics and logic and information; realms which can by definition have strict boundary conditions, and strict rules, then in such realms Truths exist.
For me, the evidence sets from many areas of enquiry, including sub atomic physics, cosmology and biology, suggest that the “Reality” within which we find ourselves is a balance of the lawful and the random, balanced in such a way that in many instances it very closely approximates hard causality at the scale of normal human perception.
The question I have repeatedly asked, which to my understanding you have not directly addressed, is:
does your assertion of universality apply just to the realms defined by the strict boundary conditions; or is it being used in the wider sense to encompass the arena of existence that encompasses us, our galaxy, this universe, and all within it.
If it is the latter, then I make the strong assertion that such a claim has been falsified beyond all reasonable doubt.
I am not making the claim that mathematics or logic is mere social construct.
I accept that both follow from the boundary conditions specified (whether one specifies such conditions in terms of information, sets, or any other systemically equivalent structure).
My initial claim, for evolution being the foundation of mathematics, is made in the sense that it seems clear, beyond all reasonable doubt, that it is the process of evolution (differential survival of variants at ever more complex emergent levels of cooperative entities), that has given rise to brains and cultures capable of conceptualising the abstract sets of boundary conditions necessary for an awareness of mathematics.
In this sense, in this reality, it seems to be foundational.
Such a claim is a very different beast from the claims of post modernism as generally portrayed in academic circles, though it comes from quite similar roots.
The intentions of mine were invariant.
The interpretations of symbols and their relationships is what has varied.
Such is often the nature of attempts at communication, particularly when high order abstractions are involved.
If one is sticking strictly to the abstract domain of mathematics, and the strong constraints present in that domain, then yes – 2 + 2 = 4 is invariant.
I make the strong claim that the moment one applies such a construct to anything real (such as counting people, or quarks), then that invariance is gone, and must be replaced by a probability statement – as per my specific example of 2 men plus 2 women (as instances of the class people), being equal to 4 with the highest probability, and also having distinct probabilities of being higher numbers – particularly in the range 5 to 12.
As to infinities, it seems that this universe within which we exist is finite within certain probability constraints, and thus doesn’t support instantiating any infinity in totality.
What evidence do you have for the continuum hypothesis applying in reality with 100% certainty???
Our best scientific theories – in respect of QM, seem to be giving us probability distributions.
How you manage to translate those to absolute certainty is beyond me – seems to be a category error in logic.
Throughout the ages, philosophers and mathematicians have made absolute claims about reality, only to have them disproven by subsequent evidence – and have them be revealed as delivering a useful approximation to something in certain contexts.
That seems to be the history of understanding.
I have no doubt that mathematics helps us build our best models of reality.
I have substantial doubt that any of those models, ever, will be 100% accurate. And they will be very useful in some situations.
And human beings have to exist in reality.
That comes with constraints on time, energy, computational ability etc.
Evolution seems to have embodied within us many levels of useful heuristics (like a love of truth) that may not necessarily be applicable to our exponentially changing reality.
Might be time to let go of absolutes, and accept best available approximations, and the humility that logically comes with such things.