Good to see you back, Tones. :up:

Is any one person ever obligated to help society? — Jerry

No, only help society if one feels the urge. Otherwise, follow the Hippocratic Oath: First, do no harm.

Many mathematicians and logicians cannot themselves be bothered to master Godel's incompleteness proof, for there isn't any payoff for doing so, and will probably content themselves with a technical understanding of the weaker version I mentioned above that is straightforward to prove and remember, and loses practically nothing — sime

Yes. Most of us are content with knowing a consistent system is incomplete. And most of us never encounter this situation.

Oh dear. But I still purr. Confusing. Disregard.

You cannot be absolutely certain that you are a cat, if you are not. — Janus

I commend dropping "absolute" altogether. — Banno

So, even though I am not a cat I can still be certain that I am? Just not absolutely certain.

The two of you are breaking new philosophical ground here. :cool:

I'm trying to say that "t" and "f" are not subsets of {t,f} because, being transcendent in the sense of the wave function, they inhabit a cloud of probability before measurement. — ucarr

My suggestion is you study the elementary theory of sets in order to use the notation accurately. Then compose your ideas accordingly.

Yes, t = transcendent fact. There's also set {f} because, through labyrinthine logic (I think), transcendence is modular to everything else, even its own attributes. Saying set {t} and set {f} are not subsets of set {t,f} is my attempt to incorporate the wave function into logical expressions. — ucarr

Are you playing the game,labyrinthine logic? The wave function is already a logical expression, subject to interpretation. This is all very mysterious. :chin:

to stretch the analogy to the breaking point, puddles do indeed make potholes for themselves to collect in when they freeze, in a sort of self-reinforcing mechanism" — Count Timothy von Icarus

Snap

But nicely said. Perhaps puddles are aliens in disguise. Clever little buggers.

One can trace a Platonism beginning in Greece, making its way through religious Christian thought and finally arriving at a humanism which retains the idea of the uncaused cause and the pure immanent identity of what presents itself to itself, but transfers these from God to mathematical idealities such as identity, pure quantitative magnitude and
extension. — Joshs

Mama say what? :yawn:

I’m careful to explain that I’m not claiming that things go into and out of existence depending on whether they’re being perceived, but that, absent an observer, whatever exists is unintelligible and meaningless as a matter of fact and principle. — Wayfarer

Very clear. When there is no observer at a site then none of the derived features of the site brought into play by a human mind exist.

Don't feel badly. I was a professor of mathematics for many years and never encountered a theorem in my area of study that was not provable in PA. Most of us don't. But there are some.

What does it feel like to be energy?

Running the hundred meter dash.

This concept of different time perceptions could indeed be an interesting scientific and metaphysical experiment, exploring the subjective nature of time and how it is perceived by individuals. It could provide valuable insights into the human experience and our understanding of time itself. — Corvus

Lots of research on Time Perception already exists. Even I have written a playful Elementary note on the subject.

I just came across one of Victor Toth's commentaries on Quora regarding Bell's results. It's the clearest I've ever read.

What are considered hidden variables in physics? Would dark matter be considered as a hidden variable?

No, dark matter has nothing to do with it.
Hidden variables arise in the context of quantum physics, in particular the famous Bell’s Theorem according to which quantum physics is nonlocal.
This is best illustrated by an example from Bell’s book, an example involving socks. Suppose you take a trip somewhere. Upon arrival in your hotel room, you notice that you have only half a pair of your favorite gray socks in your suitcase. From this you instantly infer that the other half must be left behind at home in your socks drawer. The variable representing half a pair of your gray socks was there all along, but it was “hidden” from you for whatever reason.
Now take the analogy to the quantum realm. You have a pair of correlated particles isolated from the environment, say, a pair of electrons. You measure the spin of one of the electrons and you immediately infer the outcome of a spin measurement that might be carried out on that other electron. Could it be that the spin value of the electron, just like the information about your socks, was there all along, as a “hidden variable”?
The answer is a no, for reasons that are mildly technical, but I think I can explain the essence. A spin measurement involves orienting the instrument with respect to which the spin is measured. This orientation need not be known in advance. Yet the spins of the two electrons will be correlated nonetheless. There is no classical physics analogue for this phenomenon. The point is, information in the form of “local hidden variables” that the electrons carried with themselves is not sufficient to account for the correlation between the two electrons under arbitrary orientations of the instruments used to measure them. Additional, “non-local” information is required to account for the observed correlation. Quantum physics is thus manifestly non-local, cannot be explained using hidden variables. (What is absolutely fascinating that despite this nonlocality, quantum field theory is demonstrably and strictly causal, i.e., contrary to some fictionalized accounts or even some misguided popular science explanations, quantum entanglement cannot be used to circumvent the relativistic speed limit or create a time machine. It just does not work that way, which, incidentally, is actually a Good Thing, as an acausal universe would be chaotic and unpredictable, quite possibly unstable).

I have long thought that mathematics is both invented and discovered. — Janus

That's how I see it also. When a researcher flexes their imagination and comes up with a new definition or concept, there immediately comes into existence all that can logically follow from this - and be discovered.

So, math is just a measuring and calculating tool using numbers applied to describe and predict the measurable properties of the external objects and movements. — Corvus

It might seem that way to someone who hasn't worked in the subject. But mathematicians are very imaginative people. What they have done goes far beyond what you describe. I've published a number of papers having no connection to measurement and the world of physical objects. If I had been restricted from doing so I might have become a philosopher. :cool:

The view I'm developing is that numbers and universals and the like are real, but not manifest or existent. — Wayfarer

I'm guessing most of my colleagues in the profession would agree with this. A mathematical universe is inexplicable conjecture.

But I am tempted by the possibility of mathematics being reified at the quantum levels. :chin:

To what extent can academic philosophy evolve, and at what pace?

PhD programs have as a primary purpose the launching of academic (or possibly other) careers. The grad student learns about the various topics in their subject, and then is set on a path of exploration, original research. Academic evolution thus depends largely on aspects of the subject that are studied by the senior faculty conducting the grad programs.

This tends to be a continuous process with students beginning their research under the tutelage of an experienced individual. So, frequently the next generation extends knowledge in a direction suggested by their advisors. Of course, there are exceptions. A brilliant student might come along and convince their advisor to allow them to explore unusual intellectual terrain. Jumps in knowledge may result. Or not.

The professors themselves direct intellectual traffic along various lanes, and there are any number of reasons for their choices, including possibilities of publishing an/or prestige among their colleagues.

unfortunately, universities don’t recognize the importance of a homeless, unless the homeless, as you said, finds a solution to a math professor. — Angelo Cannata

Notwithstanding movies like Good Will Hunting, I've never heard of this happening. The Indian genius, Ramanujan had a difficult time growing up, but was not homeless. He had family support, and later support from the British mathematical community.

I think the sciences are slowly moving away from the idea, exemplified by the periodic table, of pre-existing forms that reappear throughout nature. They are coming to realize that such abstractions cover over the fact that no entity pre-exists its interaction with other entities within a configuration of relations. — Joshs

Perhaps. Quantum theory is still searching for a way to understand what's happening down there. Other sciences, I'm not so sure. Intra-actions . . . who knows?

But maybe resolve it this way. Let's ask the scientists on TPF. Space, time, real? Existing? Or unreal, not existing? — tim wood

Oh great idea, go ahead, start a thread, I'll read it. — Metaphysician Undercover

Me too. But I don't see a resolution on the horizon. :chin:

I’m saying it’s an idle thought — Wayfarer

With which I agree. Welcome back from your vacation. :cool:

Space and time seem joined, and it's not clear that there is a way to separate time out and experience its passage without changes in spatial events. I mentioned the Minkowski metric in this regard, but it might be considered a man-made artifice, although it seems to work.

It does seem obvious that time's passage requires physical changes, but what is obvious to us may be superficial and misleading.

Logic is in the mind, but not [o]f it. It’s not our invention but what we are able to discover through reason. I really don’t think that the idea of a world where there are no necessary facts is even an hypothesis. — Wayfarer

You are saying you can't imagine any sort of alternate world in which the logic we enjoy would not exist. How do you know this is a universal limitation rather than a human shortcoming? A lot hinges on the definition of "possible" and our limitations thereof.

How do you reckon a world would work out, if 2 did not, in fact, equal 2, of if 9 was less than 7? — Wayfarer

If you are speaking of "worlds" in our universe, or in some parallel universe, worlds we can reckon with, then probably yes.

If you are speaking of worlds that are "possible", but not possible for us to envision, then, how could you possibly know? You wouldn't know they were possible to begin with. Are there "things" beyond our comprehension, things we don't know we don't know? How could you know? Why can you assume in some universe beyond our imagination our brand of logic must hold?

To this point, I would argue that thinking of math as a "closed," system can be misleading in this context. — Count Timothy von Icarus

I don't know how to define "closed" in this context, but I agree. With over 26,000 Wikipedia pages, and counting, mathematics continues to expand its realms, especially into abstractions and generalizations. I suppose "closed" could mean based on axiomatic set theory, which it normally is, although frequently some distance from Cantor's creations.

Further, mathematical statements are true in all possible worlds, not just in the world we've happened to experience — Wayfarer

A nice post. But I'm curious about this statement. How do you know this? :chin:

If we take space away (all three dimensions) would time exist or would it be meaningless to talk of time in such a hypothetical situation? — simplyG

The Minkowski spacetime metric seems to allow a change in the time component when space components are fixed. Or maybe not. I'm not a physicist.

Consider the intervals [0,1] or (0,1). Each requires end points, one includes its end points and the other does not. — jgill

Kindly correct me as needed, but I'm thinking both include their endpoints; in the one case the endpoints are known and identified, and in the other, unknown and unidentifiable. But whatever the status of their endpoints, both intervals. — tim wood

[0,1]={t:0<=t<=1} and (0,1)={t:0<t<1}

And finally, just for the heck of it, what is a "metaphysical object"? And what exactly is "the wave function collapse"? As an informal descriptive term, I (think I) get it. But if it's more than that, if it's a something, then what is it? — tim wood

A dimensionless point, not a pencil dot on a map. Or an infinitesimal in non-standard analysis. An object of the mind, not something that has a physical presence. IMO.

"the wave function collapse": Differential equations can have more than one solution, and a linear equation thus has all linear combinations of these solutions. Upon measurement, one discovers which of these is correct. For example, dy/dt=1 implies y=t+c, c is an arbitrary constant. Upon measurement I find that c=-.5, e.g. But this is overly simplistic and there does appear to be some weird stuff going on. In my opinion the wave function is not ontological. In fact, the Schrödinger equation in its simplest form is
dy/dt=Ky

Even pure mathematics might open unexpected doorways into reality.

↪jgill
Would you agree with me that "point in time" is at best a locution to convey informally in language an aspect of a technique useful in math, and not otherwise real? — tim wood

Would you say an interval of time is real? Consider the intervals [0,1] or (0,1). Each requires end points, one includes its end points and the other does not. What are these points? Fictions designed for the sake of argument? This makes intervals of time as suspect as their end points.

Bergson compared the unfolding of time as a tape steadily rolling off one drum and onto the other. So the word "duration" implies an infinite interval. Thus any notion of time's flow excludes finite intervals. The exact duration of an event is as much a non-real artifice as a point in time.

A photo of Zeno's arrow, frozen in flight, implies that relying on a dimensionless point excludes the recognition of the arrow's momentum. So, yes, points in space and in time's continuum are mostly mathematical objects. I say mostly since I consider them to be metaphysical "objects", outside the realm of physics - which is what many physicists concluded after reading Lynds' paper on the non-existence of points in time. Neither provable or excludable. Just fodder for endless, non-productive philosophical discussions.

Incidentally, the appearance of Lynds' article (2003) sealed the fate of Foundations of Physics Letters, which ceased publication after 2006. :cool:

Yesterday, I do swear
I saw a post that wasn't there
It wasn't there again today
I wish I wish it'd stay away

I have the opposite intuition - doesn't a physical structure require a framework of sorts? And one would think mathematical (or computational/algorithmic). — flannel jesus

When we construct a building we lay plans, then follow through physically. So those plans underlie a physical project and provide a "framework", which can be destroyed if we wish without endangering the building. Not so in a mathematical universe. Somehow the building reifies the plans and the two can not be separated. Or whatever.

The notion of "points in time" is rarely debated in science and math to the best of my knowledge. Do dimensionless points exist? This is more a philosophical issue than a mathematical one. Does a point on a ruler really exist? It certainly corresponds to a real number, but there are those who question the existence of irrationals. So, what happens to that point? It exists for some but does not for others?

Calculus is fundamental to the major branch of mathematics called analysis, founded on the idea of limits. Ordinarily, it assumes the existence of these points regardless of whether one speaks of rulers or time scales. Real analysis, the underlying structure of calculus, contains the axiom of completeness, which means points exist as viable entities. But some would say temporal points are different.

I thought of the famous debate between Einstein and the popular philosopher, Bergson, a hundred years ago. However, their issues revolved about whether time itself was independent of human experience. I'm not sure instants in time came up. But more recently a non-academic wrote a paper on time and physics in which he argues against any sort of "instant" in time. Some thought him brilliant, but others thought him a conveyor of nonsense. The latter is the more popular among physicists.

The point at infinity in complex analysis is the north pole of the Riemann sphere. No matter where you go in the plane as you move out away from the zero point the projection onto the sphere moves toward its north pole. So, in this sense, there really is a "point at infinity". :cool:

What does it really mean for the universe to be mathematical though? — Apustimelogist

The kart before the horse. What is mathematics, first? I've been a mathematician for over a half century and can not give a clear definition. There are numerous pages on Wikipedia that revolve around this question. How did math arise in human thought? Through language and observations of what we now consider logical - cause and effect - in the physical world?

Is there math without symbols? Well, yes, if one has the patience to express mathematical ideas through common language. What of the visual aspect of the subject? Well, there have been blind mathematicians who have been quite accomplished. I knew one: Larry Baggett, at the University of Colorado. So one could replace symbols with ordinary language, which seems to imply math is a substrata we contemplate by one or the other.

So, when Tegmark speaks of the mathematical universe - a creation whose structure is somehow mathematics - how can that be? Doesn't structure require a framework of sorts? And one would think physical. Maybe a collection of homeomorphic entities, that share a mathematical description, which in turn provides a uniform structure that somehow reifies.

Heady stuff that I predict will be left by the wayside of time. Or not.

And yes, people often ignorantly refer to "points" in time. But calculus usually refers to the value of a variable as some input approaches a limit - no infinities, although they're approached, and no "points in time". — tim wood

Don't be so critical. I've used "points in time" frequently in complex dynamical systems. And in complex analysis, a contour in the complex plane, z(t)=x(t)+iy(t), has a value for t=.5, e.g. And you think physicists don't use points? Do you think that limits define points, or points define limits? As for the infinite, there is indeed a "point at infinity" in complex variable theory.

So, unless you can demonstrate that it is impossible to find a better system than the use of limits, then my activity of pointing to the flaws in this system and suggesting that we find a way to change this system, is very reasonable activity — Metaphysician Undercover

This might have made a semblance of sense had you been present when my math genealogical ancestor, Karl Weierstrass, and Cauchy were pulling together the common definition of limit two hundred years ago. You could have presented them with your clearly defined objections to their work and been present for their reactions. Oh, to have been a fly on the wall. :cool:

I believe that to be able to develop technology the beings must have the physical attributes necessary to be able to create and use tools — Sir2u

Like a near fast-as-light spaceship. :roll:

There was never any outright rejection, I just stopped. I think most young people who leave the church are probably like that. As long as there isn't any pressure, they never really need to reject belief. — T Clark

Yes. That's my story, too.

Someone once said the humanoid form is universal. I think it was Orfeo Angelucci. — EnPassant

:roll:

Einstein was once asked where his laboratory was, and he simply held up a pencil. — Gnomon

:cool: Nice.

"this is the idea that mathematical truths are a byproduct of our linguistic conventions". — Gnomon

The interplay is certainly interesting.

*1. What is a Hilbert Space? :
In this blog, I aim to develop a comprehensive understanding of hilbert spaces cutting through the mathematical jargon. — Gnomon

That's nicely done. I suppose my point is that QM is all sophisticated mathematics and equally sophisticated experimental processes.

Plus the class I took was explicitly taught in the Copenhagen interpretation, and a lot of the discussions around here try to differentiate between the interpretations and, at least as I learned it, there wasn't really a way to differentiate between the interpretations — Moliere

:up: