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I guess we've all been baffled by the concept of the universe as eternal (which it must have been, since something by definition can't come out of nothing). I think eternity itself is impossible to really grasp on a deep level, because there's always a part of us that asks "When did all this actually start?" Yet it's hard to come to any other conclusion logically.

With that in mind, could it also be possible that the universe is infinite? That no matter how deep we study nature we will never reach an end, that it is infinite both in inwards (from cells to molecules to atoms to electrons and protons to quarks, etc) and outwards (infinite number of galaxies). At the face of it, this idea seems ridiculous because it's incomprehensible, but isn't it incomprehensible in the same way as eternity? Could it be logically possible? 

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I think that the best way to think of "infinity" is not as a positive concept, but as the negation of a boundary; close to how "nothing" is only the negation of something. It's a fine distinction but I think it's one worth making.

With that in mind, could it also be possible that the universe is infinite?

Let's not ask that. I would look for the answer by asking if the universe could be finite- or not.

Would the finitude of the universe lead to a contradiction?

Edited by Harrison Danneskjold
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I guess we've all been baffled by the concept of the universe as eternal (which it must have been, since something by definition can't come out of nothing). I think eternity itself is impossible to really grasp on a deep level, because there's always a part of us that asks "When did all this actually start?" Yet it's hard to come to any other conclusion logically.

I can't come up with a reason to say there is an absolute atomic singular that's a stopping point to reduction any more than there is an absolute singular starting point to existence. It's not baffling really, as long as we keep in mind existence refers to all there is in reality. it's not like there are parts of reality beyond what you can comprehend or investigate, if it really is part of reality.

 

Maybe it's weird because it's not a way of thinking you're used to, or there's a certain answer you want but there is no answer to be had. Sometimes, "that's just the way it is" is the answer.  "Why is there something rather than nothing?" Because that's how it is. You can explain how this makes sense, and all about how the concepts relate. But why it has to be so? Because that's how reality is.

Edited by Eiuol
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Be careful what you mean by "logically possible." There are lots of things that are "logically possible" that are not physically possible.

 

This is more of a scientific question than a philosophical question in mind (though, it seems that those who can't do, or don't like, the former often turn to the latter for their answers - just an observation). And as of right now, science doesn't have an absolute and complete answer.

 

Of relevance, you might consider the planck length. It is a base unit of measure derived from the speed of light with an exact mathematical (not numerical) representation. However, the physical significance of it is not known. It is believed in some physical theories derived from quantum mechanics that the planck length is the shortest measurable length, or close to it - but as quantum mechanics is a thoroughly unsettled area of physics, it's not something that's in any way settled. 

 

Physics is still trying to settle the question of fundamental particles. I'm not a physicist - I'm somewhat educated in it due to working in a physical chemistry lab, but I don't know nearly enough to give you a full and comprehensive answer. But my understanding is that the prevailing belief in physics is that there is a fundamental set of particles, and presumably a complete "theory of everything" in physics would outline what these fundamental particles are. There's no evidence to indicate or reason to suspect that there isn't a fundamental set of particles, and seeing as everything we've encountered so far is made up of constituent particles up to a point, and the existence of such particles makes a lot of theoretical sense (i.e. the existence of such particles jives well with the mathematics and the empirical evidence so far as we've been able to deduce and measure), we have reason to believe that there is a fundamental set of particles.

 

Similarly, at the macro level, we also do not know if there is a finite quantity of matter. This discussion on the physics stackexchange has some interesting information. My impression is that this is a question that we think it is likely we'll be able to answer at some point in the future, but it is not currently answerable.

 

 

On a side note, I see no reason to think this question or the concepts involved are "incomprehensible." I'd reiterate that I don't think this is really a philosophical question - there are well defined lines of scientific inquiry which may provide some answers to this question, and the scientific community - as it appears to me - certainly has some ideas about how to study the question and come to a definitive answer.

 

I'd also be very careful of your opening statement: 

 

 

I guess we've all been baffled by the concept of the universe as eternal (which it must have been, since something by definition can't come out of nothing).

 

What do you mean by "universe"? Scientifically, we have no absolute idea about what happened before the big bang. Some theories posit some notions of what might have come before the big bang based on various sets of empirical evidence or mathematical theory. There may indeed be something which is eternal and has always existed, but by no means is that necessarily our universe as we conventionally think of it (constituting all matter and energy).

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Be careful what you mean by "logically possible." There are lots of things that are "logically possible" that are not physically possible.

 

This is more of a scientific question than a philosophical question in mind (though, it seems that those who can't do, or don't like, the former often turn to the latter for their answers - just an observation). And as of right now, science doesn't have an absolute and complete answer.

Science can't say if there is in principle a point where reality bottoms out. Science is about observations, so it's important, but this is the sort of question about the very nature of what we've observed. How would you, in principle, know that you've hit the bottom? You'd need some philosophical principle. Whatever those some ideas are, they're probably just philosophical principles anyway: how to study the "smallest particle", and how in principle there is a bottom. If science can answer it, you're pre-supposing there is a bottom and that it is possible to find. You didn't say why or how it's possible that there is no bottom, so you left out how science can find the other answer. I'm not saying science can't help or isn't required to inform an answer, I'm only saying that you need more than the scientific method.

 

Regarding the big bang, you're right that something may be eternal, but you can still say "existence always has been around",

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Would this stand using an ontologically based logic?

Explain what you mean by this?

 

To back up my statement, there are plenty of logical rulesets that could lead to results that don't necessarily have a physical, reality-based interpretation. For example, in mathematics, one could meaningfully work with a spatial coordinate system extending out to 27 coordinates or dimensions - or infinitely many dimensions. That doesn't mean there's a physical interpretation of that based in reality. One could posit any number of logically sound ideas based off of reasonable axioms without having any meaning in reality.

 

 

Science can't say if there is in principle a point where reality bottoms out. Science is about observations, so it's important, but this is the sort of question about the very nature of what we've observed. How would you, in principle, know that you've hit the bottom? ... I'm only saying that you need more than the scientific method.

 

 

I disagree with you on this point. Science is about observations, in part, but it's also about theory and about logic. In fact, a very large portion of theoretical science is not based entirely on observation. Many particles that we're aware of now were predicted by mathematics long before they were ever observable (e.g. the higgs-boson), and some are still not observable. 

 

If we were to attain a sound "theory of everything" which fully explained all observations and stood up to all tests, and it predicted no more-fundamental particles than what were already discovered or known of, we would have no reason to expect there to be no more-fundamental particles than we had seen. That isn't to say one wouldn't continually test the theory.

 

I don't think that philosophy is necessary to answer this question. This is a question very firmly based in mathematics and science, and even now we're working on theories that might postulate a finite minimum length (see my first post in this thread, which mentioned one theory/interpretation coming out of quantum mechanics that posits that the planck length is, at the least, within a factor of ten of the "minimum" size of a thing in our universe.)

 

This isn't to say that philosophy doesn't have any place in it, or that philosophical logical can't perhaps offer some guidance. Just that I believe the absolute answer to this question comes down to science. 

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Scientifically, we have no absolute idea about what happened before the big bang.

 

 

I have enjoyed reading your other comments but this one struck me as not fully formed. Spacetime's time vectorfield cannot be extended earlier than the big bang and so time has no meaning "before" the big bang. Time is part of spacetime so asking what happened before spacetime existed does not make sense. What is more, the big bang is forever hidden behind an impenetrable thermal wall and can never be observed. We can only theorize about what happened in the early moments of the universe and extrapolate based on what can presently be observed. This is why it is not possible to have a scientific, absolute idea about what happened in the early moments of the universe.

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I disagree with you on this point. Science is about observations, in part, but it's also about theory and about logic. In fact, a very large portion of theoretical science is not based entirely on observation. Many particles that we're aware of now were predicted by mathematics long before they were ever observable (e.g. the higgs-boson), and some are still not observable.

Let me put it this way. I do not disagree about you regarding what science provides to this question. The only issue I had is the way it sounds like philosophy plays no role for an answer here, that it's not a philosophical question. I don't understand that. If you postulate a minimum length, you need some philosophy to suggest if it is possible, or you question your underlying philosophy if it is apparently a contradiction. On that level, it's not just physics, or theoretical physics. It turns into philosophy of physics. Abstract more, it turns into metaphysics.

So, the absolute answer coming from science doesn't make sense. The question pertains to metaphysics. It's not about only theoretical physics. It's different because it's more abstract. Suppose there is a smallest particle - what does that entail about reality? I'm thinking that you'd only know you reached the bottom if you had a metaphysics where it makes any sense. I am aware of planck lengths and all that, but I think that point of discussion is primarily philosophy. It isn't strictly the scientific method to demonstrate that the planck length can't just be reduced to mini-planck lengths.

Your first post just seems awfully close to "philosophy has a place, but science gives us real answers".

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I have enjoyed reading your other comments but this one struck me as not fully formed. Spacetime's time vectorfield cannot be extended earlier than the big bang and so time has no meaning "before" the big bang. Time is part of spacetime so asking what happened before spacetime existed does not make sense. What is more, the big bang is forever hidden behind an impenetrable thermal wall and can never be observed. We can only theorize about what happened in the early moments of the universe and extrapolate based on what can presently be observed. This is why it is not possible to have a scientific, absolute idea about what happened in the early moments of the universe.

 

If you were struck by the impression that my ideas on this topic are not fully formed, you would be entirely correct. :P

 

The impression I've gotten from my reading and my discussion with other physicists is that there are theories with some basis that posit what might've existed before the big bang. I do not have a strong foundation of knowledge in higher level physics, so anything I say about the topic should be taken with literally the largest grain of salt you can find, lol.

 

 

If you postulate a minimum length, you need some philosophy to suggest if it is possible, or you question your underlying philosophy if it is apparently a contradiction. On that level, it's not just physics, or theoretical physics. It turns into philosophy of physics. Abstract more, it turns into metaphysics.

So, the absolute answer coming from science doesn't make sense. The question pertains to metaphysics. It's not about only theoretical physics. It's different because it's more abstract. Suppose there is a smallest particle - what does that entail about reality? I'm thinking that you'd only know you reached the bottom if you had a metaphysics where it makes any sense. I am aware of planck lengths and all that, but I think that point of discussion is primarily philosophy. It isn't strictly the scientific method to demonstrate that the planck length can't just be reduced to mini-planck lengths.

 

If there were empirical evidence and mathematical theory indicating that there is no way for a discrete unit, a fundamental particle, to exist below the planck length, how would you respond to that? 

 

 I'm trying to come up with a more immediately intuitive way of looking at our disagreement - is there any way to frame this discussion where, rather than discussing a minimum length, we're discussing something more intuitive where it's easier to see our disagreement? Because I'm not entirely sure at what exact point we're in disagreement here.

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ludicious

 

You do not need a different example, or need to discuss something more intuitive.

 

You both are correct but both might claim more than your due territory.

 

There is a passing of the baton from metaphysics and epistemology to the special sciences.  The sciences must play within the fundamental principles discovered through philosophy, metaphysics and epistemology, but philosophy itself has nothing to say about the special sciences (as LP reminds us so often).

 

That said, if the philosophy is BAD, the special sciences will be able to correct it.

 

Be careful both of you.... the dance is delicate, many a philosopher has stepped on the toes of scientists (invalidly so) on purpose, and many a scientist has stepped on the toes of philosophers with naïve blindness.

 

...........

 

As for the specific example of a minimum length, such a thing is not within the realm of philosophy.  This is very similar to the question (before the discovery of atoms) of whether solid matter was infinitely divisible (this is to be contrasted with the ridiculous and impossible notion that matter is already infinitely divided).  Philosophy is primarily concerned with wide abstractions and integrations to generate fundamental principles generally applicable to ALL existents and all knowledge and all thought rather than specific knowledge requiring specific experiment and observations regarding some specific existents or aspects thereof.  the nature of space in particular whether or not there is a minimum, i.e. a minimum relationship (if space is seen as a relationship and not reified) must be confirmed or denied through observation.  There is no overarching general philosophic principle which can weigh in on the matter.

Edited by StrictlyLogical
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Explain what you mean by this?

 

To back up my statement, there are plenty of logical rulesets that could lead to results that don't necessarily have a physical, reality-based interpretation. For example, in mathematics, one could meaningfully work with a spatial coordinate system extending out to 27 coordinates or dimensions - or infinitely many dimensions. That doesn't mean there's a physical interpretation of that based in reality. One could posit any number of logically sound ideas based off of reasonable axioms without having any meaning in reality.

 

Using your example, the rift between physically possible and the spatial model is severed by creating the model without regard to the other. Baking a cake requires mixing ingredients together and heating for a period of time. If I take hamburger and breadcrumbs and eggs, mix them together and toss into the oven, it is not cake that is going to come out, rather you end up with meatloaf. Yet, I've mixed ingredients together and baked them.

 

Arbitrarily mapping out 27 coordinates per logical rule sets would lead to results of those coordinates following the rule sets, no more. Both the coordinates and the rule sets would need to have been derived from reality to have correlations to realty

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Using your example, the rift between physically possible and the spatial model is severed by creating the model without regard to the other. Baking a cake requires mixing ingredients together and heating for a period of time. If I take hamburger and breadcrumbs and eggs, mix them together and toss into the oven, it is not cake that is going to come out, rather you end up with meatloaf. Yet, I've mixed ingredients together and baked them.

 

Arbitrarily mapping out 27 coordinates per logical rule sets would lead to results of those coordinates following the rule sets, no more. Both the coordinates and the rule sets would need to have been derived from reality to have correlations to realty

 

That makes sense. I wasn't sure if you asked the question as a criticism of the points I was making, or if it was just a clarifying question.

 

If your set of logical rules is inherently limited to only what is physically possible, then my comment about the "logically possible" and the "physically possible" would not make sense. As it stands, we can conceive of a mathematically logical universe that is infinite and still possesses all or most of the currently observable properties of our universe and fits all or most current empirical evidence. However, by the same token, we can conceive of a mathematically logical universe that is finite and satisfies the same criteria. 

 

(Note, when I say "mathematically logical," I don't mean some special case of "logic" - there are multiple logical rulesets that could be laid out by obeying different sets of mathematical axioms, and as far as I understand it, the philosophical study of logic is no different. The two are inherently related.)

 

If we constrain our logical ruleset to necessarily only coinciding with the physical reality of our world, such a ruleset would, as yet, have no answer to the OP's question, just as science does not - because we are limited in our ability to perceive certain things, whether due to the constraints of our instruments or the constraints of reality (e.g. due to the nature of light, there are parts of the universe which could exist without us currently, or ever, actually being able to observe them due to the light of those parts of the universe not having reached our corner of the cosmos). For the reasons behind this, I would point to StrictlyLogical's post above. 

Edited by Iudicious
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If there were empirical evidence and mathematical theory indicating that there is no way for a discrete unit, a fundamental particle, to exist below the planck length, how would you respond to that? 

 

 I'm trying to come up with a more immediately intuitive way of looking at our disagreement - is there any way to frame this discussion where, rather than discussing a minimum length, we're discussing something more intuitive where it's easier to see our disagreement? Because I'm not entirely sure at what exact point we're in disagreement here.

I'd say similar things. I'd ask how it is they know that, I'd say whatever response I get is a philosophical answer, or that the mathematical theory in question relies upon a philosophical foundation which allowed them to arrive at answer of that kind. Often, math has theories, numerous ones, but on its own can't say which axioms to accept, and answering that is basically philosophy of math. Science is that way too, so we'd need some philosophical grounding to say that, yes, a fundamental particle, or planck length makes sense, and we can't reduce more. Since right now there is no answer, finding out an answer takes a mixture of all these things, while metaphysics grounds which answers we can look for. But that's not to say you can't question your metaphysics if science shows all sorts of weird things.

 

I guess I only disagree with this line: "I don't think that philosophy is necessary to answer this question."

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I guess I only disagree with this line: "I don't think that philosophy is necessary to answer this question."

 

Based on the discussion thus far, I would agree with your disagreement. I was, beyond any shadow of a doubt, being rather rash when I included that line. That rashness comes from a tendency that I've seen in some places to try to use a great deal of philosophizing to answer questions that require at least some science. 

 

There is certainly a great deal of philosophy underpinning science, and to rely on our mode of scientific inquiry to answer any question at all requires a significant philosophical underpinning, whether the scientists carrying out that process of scientific inquiry are aware of that philosophical foundation or not. What I should have said was this:

 

Philosophy alone cannot provide us with an answer to this question - there is good reason to believe that the universe, on a macroscopic scale, may be infinite or finite, and significant evidence and mathematical theory supporting either conclusion, and, furthermore, there is significant evidence supporting the notion that there is a finite, discrete unit at which one cannot meaningfully get smaller (i.e. there are no more fundamental units than some specific set which has not yet been fully discovered). 

 

Philosophy, however, certainly plays a part here, and can offer guidance in our lines of inquiry in this question, and certainly there may be philosophical implications of whatever science may eventually ascertain to be the reality of our universe, whether finite or infinite. 

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That makes sense. I wasn't sure if you asked the question as a criticism of the points I was making, or if it was just a clarifying question.

Your answer helped to clarify it for me.

If your set of logical rules is inherently limited to only what is physically possible, then my comment about the "logically possible" and the "physically possible" would not make sense. As it stands, we can conceive of a mathematically logical universe that is infinite and still possesses all or most of the currently observable properties of our universe and fits all or most current empirical evidence. However, by the same token, we can conceive of a mathematically logical universe that is finite and satisfies the same criteria.

If the use of infinity is delimited to math, what could be used in its place as a predicate of the universe here? What aspect of existence, or the universe, is being sought to identify?

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Your answer helped to clarify it for me.

If the use of infinity is delimited to math, what could be used in its place as a predicate of the universe here? What aspect of existence, or the universe, is being sought to identify?

 

So, here's my issue with what you linked - and it's the same issue that caused me to speak so rashly with regards to the place of philosophy in scientific issues: all of the quotes on that page discount the notion of something infinite existing in reality and speak of mathematical infinity and the meaning of infinity in the context of mathematics or science - but not a single one of the people quoted has any credentials, training, or education to back up what they're saying. They're talking out of their ass based on their personal experience and beliefs. I think Peikoff and Binswanger and Rand are/were all very intelligent people - but Einstein was a very intelligent person too, and I still wouldn't go to him for his opinions on psychology or ecology or 17th century paintings, because these were outside of his area of expertise. 

 

Objectively speaking, there's no solid reason as yet to believe that the universe is finite. There are theories positing a finite, discrete unit of existence, but these theories are not among the "settled" body of science - they are still very much in the works, and we're still working on empirical evidence or a more complete set of theories. We don't actually know if there's a finite amount of mass and energy in the universe. The universe could be infinite, expanding outward to an arbitrary distance and maintaining the same mass density thoughout. It could have a finite volume and a finite mass. It could have infinite volume and finite mass. There are theories that propose all of these things, and because there's no way to fully observe the universe, there's no way as of right now to be certain - we have no empirical evidence or strongly supported mathematical theory that would settle the question right now.

 

Which means, as of right now, there is very real reason to believe that the universe may be infinitely differentiable (i.e. can be broken up into continuously smaller units, infinitely) and that it may contain infinite mass, energy, and/or space.

 

And that besides, infinity isn't just some mathematically useful notion. In the realm of permutations, there really are infinitely many permutations in reality of any number of things. In the realm of scientific theory, infinity is mathematically useful but corresponds to physical realities: for example the notion of a Hilbert space is used extensively in quantum mechanics, and the theory would not be complete without it. It's used extensively in thermodynamics, as well as a number of other physical theories. 

 

 

As for the rest of your question, I'm not sure how to answer - what do you mean when you ask what could be used in place of the concept of infinity as a predicate of the universe? I'm not entirely sure what you're asking.

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I did predicate my suggestion with an if.

 

As to more accredited voices on mathematics, you might consider Dr. Corvini and Robert Knapp for their offerings.

 

Aside from the ability to extend and divide number, to what would infinite apply with out it being a non-specified amount, or a larger than or a smaller than, yet unspecified amount?

 

Eternity, as posited in the OP is similar. Eternity is how much time? To measure a span of time, you need two reference points. How much time as passed between now and then.

To measure length, think of the joke about the one-armed fisherman that caught a fish this long (holding his arm out with one palm as an indicator).

 

Every volume is either specific, or unspecified. Every distance is either specific or unspecified. If we disregard this as a consideration, we are back to arbitrarily assigning 27 coordinates following as set of rules, valid only within the model of consideration.

 

In an All S is P statement, to state the universe is infinite sets the universe as the subject and infinite as a predicate. What is the identity of infinity?

 

Trying to determine what an infinite amount (quantity) of mass or distance means, misdirects the focus.

 

To add one more example from Knapps book "Mathematics is about the world", He is discussing the Axiom of Archimedes. He states that mathematics cannot offer a proof of the axiom with a footnote. The footnote indicates that one does find proofs in mathematical literature, but that it has already been introduced in some other form—as per the example of Euclid including it as Definition 4 in Book V. This is the basis in Aristotle's Physics, as part of his argument against the existence of actual infinite magnitudes.

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So, here's my issue with what you linked - and it's the same issue that caused me to speak so rashly with regards to the place of philosophy in scientific issues: all of the quotes on that page discount the notion of something infinite existing in reality and speak of mathematical infinity and the meaning of infinity in the context of mathematics or science - but not a single one of the people quoted has any credentials, training, or education to back up what they're saying. They're talking out of their ass based on their personal experience and beliefs. I think Peikoff and Binswanger and Rand are/were all very intelligent people - but Einstein was a very intelligent person too, and I still wouldn't go to him for his opinions on psychology or ecology or 17th century paintings, because these were outside of his area of expertise.

I remember reading the Rand quote there recently. She had no beef at all with mathematical infinity, and said so. The point is that trying to make infinity a real thing doesn't make sense. Part of why it makes no sense is everything before that quote.

There is no tangible or platonic "infinity" to find, any more than there is a god to find (unless you want to be permanently agnostic about knowing until science says so, even about god). This is what I was saying by metaphysics grounding the answers we can find, where philosophy would say you'll be able to find a platonic planck length embedded within reality, or if a planck length is a tool. Some mathematicians really take a platonic view, others don't. This is a normal philosophy thing, we can, right now, know by metaphysics that a concrete fundamental is sensible or not. So there are reasons to believe definitively one way or the other. And if through further investigations your metaphysics seems to be plain bizarre, it's worth re-evaluating your metaphysics. The possibility of being wrong isn't enough to say "there is no solid reason as of yet". Besides, the question here is about evidence for a general answer - not energy/mass/particles/etc.

"infinity is mathematically useful but corresponds to physical realities:"

Right, but that doesn't mean there is literally infinity. A Hilbert space isn't literally space. Vector spaces aren't literally spaces. Dimensions in math aren't literally spatial dimensions. All those things are ways to understand and represent reality. They help to solve abstract problems. It's a whole philosophical argument to say what these mathematical objects actually are, in terms of metaphysics.

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Isn't this the analytic-synthetic dichotomy at work? A stake is being driven between concepts and their referents.

It's another way of saying "valid but not necessarily sound". The dichotomy is deeper, that there's not only a difference, but that there's a wedge in between concepts and referents, between logic and observation.

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You're imagining that the universe itself exists within a universe where time exists. The universe itself doesn't have the attributes of the things within it such as time and space. It's hard, but try to only ever think oft he universe from within the universe, you cannot imagine it as this ball of galaxies with you standing outside of it looking in. It's important to note that Ayn Rand, as far as I know, thought that the universe was an eternal, finite plenum. That there was no part of the universe where nothing was (true metaphysically empty space - void of any physical existence - does not exist, because where there isn't anything, there isn't anything, so it's nothing, and nothing does not exist), and that it had always been.

 

Honestly, it used to bother me, until I realized it doesn't matter. I'd rather focus on my career and life. This stuff is fun to speculate about but it's not really important. The important thing philosophically is that you know existence exists, and you know that time exists, you know that within the universe things have a beginning and an end, and that A is A meaning things are what they are and have definite existence, within the universe. Your understanding of reality, your metaphysics, is sufficient to live.

 

If you'd like to learn more about the philosophy of physics though, I'd highly recommend listening to David Harriman. He helped clarify a lot in my mind on questions of space and time.

 

Here is a taste.

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How did we discover mathematical infinity? We didn't count to it; the concept of "infinity" MEANS that you can never do that. Rather, we inferred it from the knowledge that every single number is 1 smaller than the next number. Whether it's 5 or 50000000000000, you can always add 1.

Since you can always add 1 to any number, for there to be a biggest number would mean that numbers!=numbers; A!=A. That's not possible (the very bottom line at all times and in all things is that A=A), so you will never be able to count all of the numbers.

An infinite quantity is an unbounded quantity.

We can't positively prove infinity, ever, about anything, because that would be trying to prove a negative. What we can do (as we did in mathematics) is to prove that the possibility of any upper bound whatsoever, on whatever, is not possible.

How?

For space, wherever I'm at I can always move to a spot that's one foot in any direction. For time, whatever moment I remember, there was always a moment before and a moment afterwards.

These do not constitute a refutation of any boundaries to space and time, within our modern context of knowledge, because we (by which I refer to people far smarter than me) have realized that these facts could be logically consistent with certain geometrically bizarre boundaries, if they were large enough. These facts do, however, demonstrate the sort of thing we need to look for in order to sensibly reason about infinities.

If you want that knowledge then you've got to think about it recursively.

Edited by Harrison Danneskjold
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