Citation needed :-)
>(how can you perform an operation like 1+1 without time?).
As a long tautology -- all steps exist in advance. There is no "movement" in mathematics. 1+1 is already 2.
Same way in e.g. a holographic universe could have all developments present at the "same time" (or rather, in a timeless state). Where time is just the perception of focusing on one or the other pre-existing point (with no actual process of change from one to the other).
1+1 is a dimensionless computation and hence has no notion of time.
To use time, we need to add it explicitly as a dimension to the numbers used.
We generally use SI units to abbreviate the dimensions. I.e. s = seconds, m = meters.
Now 1s + 1s = 2s. There we have used second as the dimension of the unit.
We can do the same thing for lengths. 1m + 1m = 2m.
Speed is generally expressed as a dimension, that when multiplied by time, gives us the length travelled.
speed * time = distance
or
speed = distance/time
If we set time = 1 s, distance 1=m, we can deduce the unit of speed as.
speed = 1 m / 1 s = 1 m/s
Hence we get the dimension of speed, m/s ,or meters per second.
We can now ask time related questions like "How long - how many seconds - does it take to travel 6 meters with speed 3 m/s". We can just observed the dimensions to get the correct calculation:
6m / (3m/s) = (2/1s^-1) = 2s
So that is how mathematics and time generally interact. Time is used like a dimension, just like distance or weight.
I'm not sure if this is what you meant or something else.
Below is written from a specific pedestrian view of mathematics, and is partly arm-chair philosophy.
Similarly to one playing a specific piece from sheet music, the sheet music is an invariant of time, while the act of performance, is, yes, very much time related.
I think what you mean is the act of performing the computation 1+1 always takes some time, what mathematically is generally understood as relevant is the relationship 1+1 itself, which does not really change.
We can change the numbers so we don't know the result beforehand, but that does not mean the end result would change - it does not, as long as the rules don't change.
The relationship of the diameter of a circle and the the circumference is always pi. We can take any number of of circumferences and diameters, and given one or another is unknown, we may need to do some computations to know for example the circumference of a wheel with a diameter of 2.
While we need to perform the computation pi * 2, the result itself actually never changes, and is irrelevant of time -an eternal truth so to speak, crystallized forever in the fundamental logical nature of our reality.
Or, another example, if we take a right triangle, say with the Egyptian famous example with side lengths of 3,4, and 5, it gives us many more invariants that don't change no matter how many times you perform any computation.
Given a right triangle with non-hypotenuse sides 3 and 4, the hypotenuse is always a length of 5. No matter how many times we perform the computation, for example, sqrt(3*3 + 4*4) - the end result is something timeless.
Pythagorean theorem holds for all planar right triangles, for all eternity, and everywhere, Acapulco or for example on a some desolate rock circling a star in Magellanic Clouds 2 million years ago.
While "Pythagorean theorem" is just a name given to a specific logical relationship, the logical relationships described by the theorem do not change. They are eternal.
Discovering Pythagorean theorem certainly took time.
One view of mathematics is that it's just a way to navigate a vast logical landscape of logical relations that are invariant of time or space. That the act of "performing" mathematics is navigating a static landscape crystallized beyond space and time, into a fundament.
Others view mathematics as just a logical game. Never the less, even there with fixed rules, specific input will always result in specific output.
Nothing in math has time or duration. 1+1 = 2 = area of a triangle with base and height 2 = 362/181 = -2(-1). These are timeless relations. If 2 has no duration, none of the others do. They are just statements of relations.
In fact ontic structural realism (some versions) says there is only these abstract relations and everything can be reduced to mathematical relations. Spacetime, consciousness, subjecive experience - all of it are just timeless relations like numbers in a table.
It's a tempting view due to its simplicity and the importance of math. But it is extreme, but so is every other theory or metaphysics trying to explain the same stuff!
It can be that this is even impossible since as with equations the words of language are static and can only refer to the perception of time without capacity to describe it.
What no one has done is understand consciousness or reduce biology to physics much at all. The math behind the above is extremely well known and ironed out. That is why some physicists now put the challenge on biologists and psychologists. The personal or subjective flow of time we all experience is theorized to be recreatable from this (timeless) framework. After all, all you have access to is a single snapshot and memories. Sensorial data and brain states. Registering and firing of nerves and neurons. Momenta of particles...why not? Just particle momementa in a table in some kind of sequential form (e.g. worldines in 4D spacetime block universe).
I completely agree this may end up being an impossible challenge, or that it's later shown to be misguided. But there are accomplished philosophers and physicists behind it, or at least sympathetic to it.
I.e. a physical model is like a set of frames for a movie with particular relations between frames. But it does not tell why if we see the movie we perceive the motion and not, for example the whole movie at once. Or why the perception is across time and not space.
And since these being discussed in various forms since at least Parmenides and Buddha I doubt this will be resolved any time soon.
What do you mean by this?
When I write x(t) = x0 + (dx/dt) * t + (d2x/d2t) * t^2, time is right there in the equation.
In relativity, it is even self-evidently different from the 3 spatial dimensions: s = sqrt(x^2 + y^2 + z^2 - (ct)^2).
There is even a somewhat intuitive interpretation for the special-ness of the time dimension: all matter is constantly moving along the ct axis of space time at speed c, unless acted on by some force, trying to follow the shortest distance between the past and the future (so the trajectory bends towards the center of mass of any other matter it finds on the way).
The fact that our mathematical models generally seem to allow movement backwards and forwards in time is in stark opposition to observed physics. This obviously suggests that the models are wrong, and there is something about time we are not capturing well enough. Relativity does somewhat fix this, by essentially postulating that everything is moving with speed c in space-time, but that is a somewhat unsatisfying explanation, since it seems natural to ask why.
I am not saying that time and space are the same in physical equations. The point is that they do not describe the perception of time flow, not that there is no difference between time and space.
As I wrote the structure of equations (equations are parabolic or hyperbolic, not elliptical, the sign of the time dimension in the metric tensor is opposite to the space coordinates) reflects that one can predict across time but not space. I.e. the equations reflects that from a picture of a room one can tell what will happen in one hour or what did happen one hour ago. Shadows from the Sun will move, a sleeping cat will not be there, but things will be mostly the same. But try to tell what is in the rest of the room from a one-hour long video of the wall. It is not possible.
But this difference tells nothing about perception of “now” or the time flow.
In a sense the equations reflects how memory operates. We remember a sequence of events and we can focus on a particular moment or select events in an arbitrary order similar how we can select a point or points on the world line of a particle. But the memory does not have “now” and so the equations reflecting the notion of the world line tell nothing about the time flow.
Well, you've eliminated one step: you can't just look at particle positions, you need to measure change in particle positions in relation to X if you are to explain the world. Simply saying "static state 1", "static state 2" doesn't explain anything, unless you have some kind of relation between state 1 and state 2.
In the end, the very basic purpose of physics is to be able to tell, given state 1, what state 2 will be.
Totally. But I would say your idea of predicting is still unnecessarily involving a "flow of time"; of things "becoming" or changing in themselves. Relations can hold without this notion of change or "becoming". A carpet with a regular pattern does not evolve in time, yet it has patterns. The patterns could be analogues to the laws of physics. The pattern tells you what can be in adjacent portions of the carpet, just like the laws of physics take input and tell you what will or can happen next. The carpet never changes though. You can imagine a separate observer tracing out a line in the pattern from one section to another and seeing it "change", but again the carpet never changed.
Fitting consciousness into this model is very hard I admit. There would be no outside observer tracing out lines in the carpet seeing them change. Instead, the flow of time (the feeling of us persisting from one moment to the next; not being entirely new objects at each instantaneous slice) would have to be a conscious illusion. But so is redness and warmth I could say. Maybe a brainstate is just your memories. And having a sequence of memories gives the illusion you existed prior to this instant. And this process* is done at every instant.
*Process as in timeless patterns on a carpet or timeless laws of physics. This process does not "take time", it's just action of particles behaving the laws of physics - in a timeless/carpet sense. Consiousness is given at every instant by the relation of particles, like the imagine on a movie screen is given at each instant from the photons of the projector. It's just that our screen "consciousness" feels persistent due to having memories.
If anyone wants a more cogent explanation read Tegmark, Barbour, and Harvey Brown. Here is Harvey https://youtu.be/CA-YsWXRSHU
The problem is even worse, because without some time-like process, that geometry can't even be observed. The problem here isn't mathematics, it's the practical aspects of the universe.
> timeless universe within which I sit with the illusion of time
I'm not sure what the definition of illusion is here, but the universe does have an evolving state and that's not an illusion according to any meaning of the word I can think of. Time as a general measure of the speed of this evolution is not a human concept either, there is no difference in the way living matter is subjected to time vs how "dead" matter is.
Where there's room for interpretation in my opinion is whether time is a real dimension, or if it's just a dimension-like phenomenon. It may in the end well boil down to something like "time is just the local tick rate of matter", which is a position that may only be provable if we could somehow discern whether the past actually exists physically (as opposed to being destroyed as the universe's state evolves like an array of cellular automata would).
There's a bouncy ball in a box with pressure sensing walls. Each wall feeds the pressure data to HDD physical storage. The ball is released from the top wall at some random angle toward the bottom. The storage devices spin at 100prm and in sync.
Looking at the data from the 6 storage devices, I would see the first entry is from wall_ceiling, recorded at plate rotational angle 0pi. Then the next entry in the collection is from wall_floor say at 1pi. Then wall_left at 1.5pi then wall_ceiling again at 2.1pi, then wall_right at 3pi.
The ball-sensor-storage system alone never needed a "flow of time". The data printout at the end is just a piece of paper with a few columns of numbers. A function is also a column of numbers. And so is a circle or any other polygon.
You may think the printout went from 1 entry to 2 entries to 3 entries...that surely was a flow of time. No. A flow of time is the subjective feeling of a continuous you "flowing" through time. It is entirely internal. The printout system does not have a flow of time. It has time, but again, Barbour is arguing time is extra baggage which is unneeded when you have the motion of the particles - as above. It adds nothing. At each step the printout is entirely given by the behavior of the other particles in the system. Why not our brain states too? When you see "blue" you are seeing 480nm wavelength photons hitting your retina and cones. When you feel continuous, each brain state of yours is kind of a running average of your previous brain states (memories). Or something like that. Maybe some kind of recursive build up of memories at each instant.
But you say you must "flow" even for brief moments to perceive something. Even if it's in discrete steps that's fine with you. But you have to be willing to go one step further to get their arguments. There is only assemblages of particles. And somehow certain assemblages lead to self-reference and awareness, of which we call conscious experience and "flow of time".
Can a function "feel" something? Who knows. But the course of human scientific endeavor has taken us very far from where we started.
This is the part that seems suspect: it seems like motion is taken to be fundamental, but time is not. This seems just as arbitrary as taking time as fundamental, and motion as derived (simply the totality of positions at different points in time).
But overall, given the position and momentum of a particle at step 1, can you predict its position at step 6 without knowing some quantity that is equivalent to the length of time between step 1 and step 6? And if you do require such a quantity, does it matter whether we use time or something else?
> A flow of time is the subjective feeling of a continuous you "flowing" through time.
Sure, but there is much more to time than this human subjective experience. A lot of our knowledge of physics relies on the notion of time to actually function. It's true that we already know that some of our fundamental physical theories are flawed (since they are currently incompatible with each other), and perhaps removing time from them COULD be a necessity for unifying them, but this is far from a given.
I very much doubt you could reformulate special relativity without ending up with a concept that is completely isomorphic to what we think of as time.
Note that a physical representation of a continuous "you" is a much more dubious concept than time itself, and I think it is one that is anyway incompatible with most physical theories/interpretations, as arguments like the Ship of Theseus or teleportation have shown forever.
In particular it's the casual anthropocentrism that doesn't sit right with me, but it's so much more. You're mostly concerned with human perception and philosophical aspects that have roots in human feelings. If your main objection to the existence of time is that it doesn't fit in with your definition of consciousness, that means we're not even remotely talking about the same thing. We might use overlapping words, but that's it.
Consciousness is not a scientific concept.
Penrose did a lot of damage trying to legitimize it with his quantum woo, but it's fundamentally incompatible with scientific considerations outside of psychology.
Of course there is state change. Of course the sun will rise tomorrow. Of course a state machine will evolve deterministically based on some inputs.
Those are all described mathematically. But the mathematical laws of physics alone must then be interpreted in some way to describe reality. I mean look at the debate over which interpretation to apply to the same maths of QM for an example. Physics is not mathematics, it applies explanations.
And what these physicists above argue is that the math of the laws of physics does not necessitate an ontological, mind-independent flow of time to the universe. That we do experience a flow of time is not under question. We do. What is under question is that since the laws of physics (i.e. the math) works without an ontological flow of time (note: flow of time is a different idea than "time" or "arrow of time", neither of which I am arguing about here. We are only talking about the flow of time), why and how do we experience/perceive one?. Well, they say, through the sequentialism of memories and physical, sensorial data. Nothing woo. And if you are worried about physicists talking about experience - don't we experience a photon registering on a detector or a magnet deflecting electrons up vs down? Those come through empirical experiences. The flow of time would be an experience in a similar form as redness is. It has an empirical origin (e.g. photons on our cones) that leads to an experience.
So no one is denying the flow of time as experience. Just like you can't deny "redness" or "hotness". They are experienced by us, but some also think we can explain them in naturalistic, scientific ways. And that will require a neuroscience perspective on some level. I only bring in consciousness to the extent needed, as it is intimately connected to our perceived "flow of time".
We have to keep clear, "time", "flow of time", and "arrow of time". In earlier posts I did say Barbour thinks "time" is a redundant term; needless baggage on top of just caring about particles and relations to other particles. That in turn morphed into how we can recreate the experience we all have of time-flowing if there really are only particles and forces - no magic of the mind. To explain that, I had to talk about how the flow of time might not be ontologically real. And neuroscientists and physicists will then have to show how our experience is reducible to particles and forces (or fields). A tall task but a totally naturalistic, scientific one no? No woo. Put off defining consciousness and the mind and how they emerge from particles, but allow for it to be what ultimately is responsible for experience, within a "timeless" universe.
Is this not a justifiable position? If you disagree, I would be interested where you feel it goes wrong (if you feel like sharing). If you think we can do physics without referencing to experience I would also be interested.
But in timeless universe, it should be a known final quantity, similar to 1+1, no? Because all possible halting Turing machines are in halted state there and non-halting ones can't exist there(*). So if we could peek into such universe we could sample them and get Chaitin's approximation with any accuracy desired.
(*) or, all their states exist and form a graph with closed loop, which would also be observable
(Having said that, I cannot wrap my mind around Babour's concept of an atemporal universe, either.)
An arbitrarily large, even growable, tape has more limited computational power than an infinite tape.
Not that we could have the latter, or an unconstrained version of the former, in our temporal world, of course!
[1] https://www.goodreads.com/book/show/400716.Introduction_to_t...
The word "known" is doing a lot of heavy lifting there, and bear examination for hidden assumptions. I would go with "defined". It is defined and concrete, but "known" adds a lot of questions in like who or what "knows" it. (I am skipping the question of "existence" on purpose because that's yet another question. The BusyBeaver(10,000,000)'th row of Pascal's Triangle is well-defined but whether it "exists" is yet more philosophy.)
Time does not appear in formal logic or number theory, even though it is convenient to think of a derivation or a calculation as as sequence of steps performed one at a time.
it is also very convenient to use mathematics to model a temporal process, but we can, and often do, then show what happened (or will happen) as tables, graphs or other static representations.
I know that last sentence presupposes time, but I am not even going to attempt to rewrite it in a way that avoids doing so!
In computation you generally do follow a process which follows from one step to the next. However, this doesn't necessarily imply time in any meaningful way, I think having a countable base like the natural numbers is enough (such that you can meaningfully say 'this is step 1, this is step 2, step 2 comes after step 1 in our ordering').
1+1 is a dimensionless operation.
Do you mean how it would physically happen without time? It's possible time is just an illusion that emerges from our perception of the universe.
See https://www.nature.com/articles/d41586-018-04558-7 and https://www.quantamagazine.org/does-time-really-flow-new-clu... or https://www.npr.org/2013/05/17/184775924/resetting-the-theor... for a counter argument.
It seems to me that you wouldn't be able to have the notion of "1" and "2" unless they can be separated, and to separate them you need dimensions. So for a number to be intelligible at all, it requires dimensions; but how do you get dimensionality in the first place, if your equation is specifying the universe?
But they are all at once.
I can't get my head around how anyone can generate time (whether an illusion or not) just from spatial dimensions. If I have to assume spatial dimensions to get anywhere, can I also assume time?
But, in "this world," the one we "think about" time is the path between two simultaneously existing universes. The universe a minute ago still exists. The universe a minute from now exists already. Time is how we got here and will get there.
We aren't generating it. It's always here. It only exists when it is observed. You observing it makes it exist. That's the path. You haven't observed the universe an hour from now, so it doesn't exist yet, but that's only true in your mind. It's already here. It's there. All potential universes exist simultaneously.
Time isn't an illusion. It's a path. It's like the trail from your house to mine.
For a point to get from one point to another, it needs another dimension, a line. A line needs a plane. A plane needs a cube, a cube needs time, time needs... what? The fifth dimension. Through the fifth dimension, we can create paths from one time to another.
We just aren't there yet. We haven't observed that happening, but it's there. Always was, always will be.
It is helpful, I think, to think of numbers, sets, and the like, not as anything fundamental but as tools that help us make sense of the world.
Describing patterns observed from within space-time and ending up modelling a timeless universe seems incoherent to me!
My concern is something like this: observing and modelling patterns in the universe can only take place within time. If your models derived from the observed patterns then allow you to model a universe that does not have time, then there is something fishy with the model -- you've bootstrapped a timeless universe that is only intelligible from within time.
I'm not writing that under the assumption that Kant was right, his philosophy is based on something like "inner intuitions" that are dubious and in my opinion do not match modern mathematics very well which deals with objects that go far beyond our visual imaginative capabilities and what we might check with some kind of "inner intuition" (innere Anschauung) or geometrical insights. I just wanted to point out you're not alone and if you're confused, then in an elaborate way.
[1] https://en.m.wikipedia.org/wiki/Constructivism_(philosophy_o...
Complexity (and entropy) have surely a crucial importance and can't be totally avoided but I think that mathematics precisely work at minimizing their impact.