What time is not
It is not, I think, a kind of stuff or dimension. This is for numerous reasons. Conceived of as a stuff (or dimension, if dimensions are not stuff), it would be infinitely divisible, yet nothing that is infinitely divisible can exist in reality (yet time does exist, thus it is not a stuff/dimension).
Note too that the past is potentially infinite, as is the future. But if time is a kind of stuff or dimension, then it - the stuff itself - would have to extend infinitely otherwise how could any event in it recede potentially infinitely into the past? Yet nothing that is actually infinite can exist in reality (yet time does exist, thus time is not a stuff).
Finally - and I am not suggesting these exhaust the problems - there would be no fundamental difference between past, present and future. Indeed, there would not really be such properties, only early than and later than and simultaneous-with. But future past and present are essential to time - they 'are' the fundamental temporal properties - and they are radically distinct from each other (thus, time is not a stuff).
Time, then, is not a stuff, not a dimension.
Note too that the past is potentially infinite, as is the future. But if time is a kind of stuff or dimension, then it - the stuff itself - would have to extend infinitely otherwise how could any event in it recede potentially infinitely into the past? Yet nothing that is actually infinite can exist in reality (yet time does exist, thus time is not a stuff).
Finally - and I am not suggesting these exhaust the problems - there would be no fundamental difference between past, present and future. Indeed, there would not really be such properties, only early than and later than and simultaneous-with. But future past and present are essential to time - they 'are' the fundamental temporal properties - and they are radically distinct from each other (thus, time is not a stuff).
Time, then, is not a stuff, not a dimension.
Comments (247)
And yet it can be dealt with as a dimension in mathematics and physics and predict observed results.
Quoting Bartricks
Why not? Just curious.
That's consistent with it not being a dimension.
We can note too that if it is a dimension then we would predict that it would be infinitely divisible and would extend infinitely - yet nothing can be like that, and so we now know that treating it 'as if' it were a dimension is merely useful, in much the same way as, for example, taking our sense reports at face value is often useful even though reality may not be as they represent it to be. And we would predict that there would be no fundamental difference between an event's being present, past or future (yet manifestly there is a world of difference).
Quoting John Gill
It is manifest to reason that 2 + 2 = 4 and that if a proposition is true it is not also false, and it is manifest to reason that nothing can actually extend for infinity, or be composed of an actual infinity of parts.
For example, someone who claimed there was a hotel that was full and would remain full even if half the occupants left, is someone we know a priori has said something false. There can be no such place. Yet if actual infinities are permitted, then such a hotel would be possible.
Manifest=clear or obvious to the eye or mind. And does the eye or mind unravel every detail of the universe? If there are features of reality we cannot fathom does that negate their existence? Your hotel example is not terribly convincing, IMO.
Hmm, no, just as it is clear to sight that there are colours, it is clear to reason - that is, our faculties of reason represent it to be the case - that 2 + 2 = 4 and that this:
1. P
2. Q
3. Therefore P and Q
is valid, and that no proposition that is true is also false, and that no actual infinities exist.
Obviously there is no suggestion here that our reason is an infallible guide to reality, but it is ultimately the only guide we have (such that even establishing that some of what our reason represents to be the case is not, in fact, the case would require appealing to reason).
'No' to both. But if our reason - that is, the unprejudiced reason of most of us - tells us that something is not the case, then that is excellent prima facie evidence that it is not the case.
Note too that even those who, for dogmatic reasons, insist that only the reports of the five senses give us insight into anything real, also have to appeal to reason, for it is by reason - not sense - that we recognise that our senses provide us with insight into something.
I think the important distinction would be between substances that are extended and those that are not. Time, to the extent that it is conceived of as a kind of substance, must be being conceived of as being an extended stuff. And it is extended stuff that generates the problems.
Quoting khaled
But I've argued that time is not a substance. If the case is good, then we have gotten somewhere, for now we know that thinking of it that way is a mistake.
Quoting ovdtogt
No it isn't. It 'is' the past and the future (and the present). Stop trying to be profound. Time is love on a tricycle. Time is what tomorrow needs to prevent it from being today. No, no, no.
Time is a quality, like the colour blue.
j/k, time is actually a local entropic anisotropy in the phase space of possible worlds.
By which I mean that time is more like a big ball of wibbly-wobbly, timey-wimey... stuff.
(One of these is not a joke).
My food looks really time right now
I've given it some thought and...
Motion is a very fundamental phenomenon. In fact every object in the universe is in motion relative to at least one other object. Imagine now a 100 meters race between 3 runners of differing prowess taking place in front of you. Since they're unequal in ability it's plausible and expected too that they will reach the finish line in a specific sequence 1st, 2nd and 3rd. This sequence is true and verifiable through actual first-hand experience.
The question that naturally arises is "how do we make sense of this sequence?" It isn't a spatial sequence because the race is 100 meters for all runners. Ergo, the sequence must exist in something that is not space and this domain where events can be sequenced is called time. This allows us to get a handle on what the sequence in the race means - the 1st runner took less time than the 2nd runner who took less time than the 3rd runner.
This also gives us the classical divisions of time into past, present and future. If we were to focus on the runner who comes 2nd we could call it the present; then the runner who came 1st is in the past and the runner who'll be 3rd is in the future.
Time is a domain in which events can be sequenced into past, present and future. It's very much like space where location can be sequenced into far, midway and near.
We measure time in some unit; the usual ones being seconds, minutes, hours, days, weeks, months and years etc. The most obvious unit of time and hence the first unit to measure time with is the day (24 hrs). However, the day is simply the distance the earth travels in one rotation around its axis. In other words we may reduce time (1 day) to space (the distance covered in one rotation of the earth). This model continues onto all units of time with a second being a specific distance between two marks on an analog watch.
Is it that time is just space (distance) then, with every unit of time being nothing but different distances? Recall the 100 meter race scenario above and we can see that the past, present and future (1st 2nd and 3rd) sequence isn't spatial - 100 meters for all 3 runners. The sequence is real and must be a sequence in some domain of reality and it's this domain, independent of space, we call time.
Spatial measurements are made for an object by corresponding locations of the object to the rod.
A clock is a periodic standard event (tick) generator. It allows a measurement of the interval between events of interest in terms of ticks, in addition to an ordering of those events. Time is a human concept of convenience.
Yet there is a universal speed limit - the speed of light - and speed = distance / time, so it appears that something / some mechanism within the universe must be 'time-aware' else the speed limit could not be enforced - so time seems not just a human concept - it seems to be part of nature.
Nope, they all are .. including this one.. lol
For instance, can something exist that is infinitely extended? Well, no. For instance, when a position is shown to generate an infinite regress, we consider that a damning indictment of the view. Why? Because we - most of us - recognise that actual infinities cannot exist.
Now consider that any event in the past recedes potentially infinitely further into the past. Well, if we conceive of time as a dimension then the only way it would be possible for an event to recede infinitely into the past is if time itself extends infinitely - yet as just shown, nothing can be like that. Thus time is not a dimension.
As for 'absolute' time - I am not sure what you mean.
Which is worse, bad theology or physics without the mathematics? Neither has a place in this forum.
An anti-realist with respect to time, might say that "nothing can travel faster than the speed of light" is a statement about the grammar of special relativity, rather than being a factual statement about the world.
The reason why special relativity 'concludes' that nothing can travel faster than c relative to any inertial frame of reference, is because otherwise causality would be violated by faster-than light objects moving 'backwards' in time to inform the past.
If we could make empirical sense out of this idea of causality being violated, then special relativity could not rule out the possibility of faster-than-light objects. But we cannot make empirical sense out of the idea of causal violations, since it leads to empirical contradictions. Therefore an anti-realist might argue that "faster than light travel" isn't a false proposition but a meaningless sentence. In which case "nothing travels faster than light" is a statement about the language of physics rather than a negative proposition about the world.
Thanks for reminding me. All I wanted to do was prove that time is non-spatial, despite its measurement being so and that time isn't some kind a special space. Although it's represented as a 4th dimension in modern physics it is unique enough to deserve separate treatment.
If time can be represented as a dimension then what is time for a 2-dimensional being? Would it view our 3rd spatial dimension as time just like time is a 4th dimension for us? This is a puzzle for me and is beyond my skills to answer but I'll try and explain the issue: Either time is always the fourth dimension for all possible worlds or time is an "extra" dimension added onto whatever spatial dimension a world exists in. The former scenario would mean that all worlds, regardless of how many dimensions they have would consider time always as a fourth dimension. Let's call this situation as T4 time. In the latter view a N-dimensional world would make time as N+1th dimension. Let's call this N+1 time.
If time is N+1 time then time is actually space in a higher dimension accessible to N-dimensional worlds in a limited way in that we can only go from the past, through the present, into the future.
A T4 time situation seems unlikely because there is a sense that time flows in a universal way for all worlds and each world would need to have time added onto their space as an extra dimension.
So, time is probably N+1 time but that leads to time being just space in a higher dimension. Notice though that our freedom, what we can do, is severely limited in the time dimension - we can only move forward. For instance take a ball and roll it on a table. We can roll it back so it retraces its course perfectly to it's starting position but even though the ball moved backwards in space, it can never move backwards in time for time has passed into the future between rolling the ball forwards and rolling it backwards. Another way to see this would be the impossibility of going back to 1945 to witness the end of world war 2.
Some say this irreversibility is due to entropy - that it always increases causes the arrow of time. This is where things get interesting because entropy is a physical-spatial concept. If we could reverse every spatial entity in the right way then the world could travel back in time. Imagine that every event is reversed causally (for simplicity think billiard balls) we could then retrace our step back to, say, 1945 and see the end of WW2. Doesn't this suggest that time is reversible in theory but not in practice due to entropy? This explains the unique nature of the dimension of time insofar as we're concerned.
That means, if entropy didn't behave the way it does, we could access the temporal dimension as easily as we do the 3 dimensions of space. Time then is just space in a higher dimension.
As for infinite regress, I think if we consider time as a higher dimension of space the problem disappears. Nobody will object to space being infinite and if time is simply a higher dimensional space then there should be no problem in it being infinite.
Why not? You are making it axiomatic that this cannot happen. Is your axiom true or false? How can you tell?
Which is to say no more than that you choose to use the word "dimension" in a way quite at odds with how it is used in physics.
You have invented a useless word game.
There is a great deal of empirical evidence that the speed of light is a universal constant obeyed by everything in the universe; we have been measuring it for 100s of years and we currently know it within a measurement uncertainty of 4 parts per billion.
Saying the statement "nothing travels faster than light" is about the language of physics seems to me to be equivalent to saying the statement is a natural law - the language of physics is our model of natural laws after all - so I maintain a belief that the natural laws of the universe are time-aware. This suggests time is more than just a human invention.
Quoting TheMadFool
Theoretically anyway, the ball can end up back where it started in time - you might like to read about:
https://en.wikipedia.org/wiki/Closed_timelike_curve
So if time is best modelled as a dimension, which maybe the case, it seems to be a complex, convoluted, non-linear dimension - bearing in mind the results of SR and GR.
Quoting TheMadFool
Some people do object - spacetime looks like a creation (see the BB). It's impossible to create anything infinite in size, so therefore spacetime should be finite.
I've already said. I am not 'making it' axiomatic, I am appealing to reason.
Actual infinities can't exist. Or so says the reason of virtually everyone (which is the best evidence there can ever be that something is the case).
To borrow the example of Hilbert's hotel - an example employed to illustrate the rank absurdity of supposing infinities to be actual - a hotel with infinite rooms could be full to capacity and still accept new guests (indeed, an infinity of them). And if half the guests left, it would still be full.
Now, that's true - no? And it is also absurd. If I told you I own a hotel that is full to capacity but we can take as many new guests as you like, you'd be fine with that?? You'd think "yup, there are hotels like that"?? Or would you say "er, how can your hotel be full to capacity 'and' be able to take as many new guests as I like?"
Yes, but I have not claimed that time is mistakenly thought to be space. I have claimed that time is mistakenly thought to be a dimension.
The case I have made for that is that, conceived of as a dimension, it would be infinitely divisible. Yet nothing can be infinitely divisible, and thus time is not a dimension.
So, here's argument 1:
1. If time is a dimension, then it will be infinitely divisible
2. Nothing existent can be infinitely divisible.
3. Therefore, if time is a dimension it does not exist
4. Time exists
5. Therefore, time is not a dimension
Another argument was that if time is a dimension, then it would have to extend infinitely as any event in time can recede into the past forever. Yet nothing can extend infinitely.
Argument 2:
1. If time is a dimension, then it extends to an actual infinity.
2. Nothing that exists is infinitely extended
3. Therefore, if time is a dimension, then it does not exist
4. Time exists
5. Therefore time is not a dimension
Another argument was that if time is a dimension, then there is no fundamental difference between events that are past, present and future. Past, present and future cease to be intrinsic features of time. But they are intrinsic features of time, thus time is not a dimension.
Argument 3:
1. If time is a dimension, then past, present and future are not the intrinsic temporal properties
2. Past, present and future are the intrinsic temporal properties
3. Therefore time is not a dimension
Yes, that's what I think.
Address an argument. You know, do some actual philosophy for a change.
Quoting Bartricks
(2) is false. But not just false, it is malformed. Clearly there are things that are infinity divisible.
Agreed. It is just another example of playing with words.
A substance is a bearer of properties.
And to be 'extended' is to occupy some space.
'Time' is neither. It exists, but it is not a kind of stuff and nor is it extended.
:smile:
Nor is mere assertion.
Here is your assertion: Nothing existent can be infinitely divisible
But, says I, there are things that can be infinity divided.
Like number lines. Or seven.
Hence, your assertion is wrong. But more than just wrong - your assertion shows that you choose to misuse words such as "dimension".
You are misusing the word 'misuse'.
Quoting Banno
I didn't merely assert, I appealed to self-evident truths of reason. Try it sometime.
7 is laughable example. I mean, really? 7 is a thing, is it? A thing that can be infinitely divided? And it cannot actually be infinitely divided, only potentially so. You're mistaking the number 7 with '7 things'.
Now, tell me, which premise of this argument is false and why?
1. If an infinitely divisible thing can exist, then a hotel with infinite rooms can exist
2. A hotel with infinite rooms is a hotel that can be full to capacity, yet still admit new guests - an infinite number.
3. It is impossible for there to be a hotel that is full to capacity yet can still admit new guests
4. Therefore an infinitely divisible thing cannot exist
Rationalism. It's a tried and failed philosophical method. Good for you.
So you say seven doesn't exist? Or seven cannot be divided by any other number? Or both?
Keep going; it all only serves to further my point.
Ah. Quick, change the topic...
(3) is wrong. Hilbert's hotel can always take more visitors.
And yet again, your objection shows only that you choose not to use"infinite" in the way mathematicians do,
Poor stuff.
Ah, yes, of course. Thank you oh mighty Banno. I see now. Yes. It can. Yes. Brilliant reply. I be learning much.
Sorry, I am being too subtle for you. I am saying it is not a 'thing'.
Exactly what numbers are is itself a fraught philosophical issue, but no-one apart from a PLatonist thinks they're actual things, and even they would agree that the thing that is the number 7 - the Form of 7 - is not divisible.
Hilbert was a mathematician. Hilbert's hotel was a thought experiment he devised to underline the absurdity of thinking actual infinities can exist.
But don't let that worry you. If you want actual infinities to exist, then they jolly well can.
But it is still playing with words. You define words in terms of other words, which no doubt, in turn, can be defined by other words.
If you wish to say something interesting about Time, you will need to get below the level of words; like physicists do.
Physicists are not investigating what time is. That's not a question in physics. How it behaves, yes. What it is, no. That's a philosophical question. You have to use your reason to figure out the answer.
When you watch a documentary about time, pssst, it's not about time. It'll have lots of physicists doing philosophy badly and saying weird stuff about time bending and such like - but it isn't about time, it is just about entertaining people for half an hour while they push fish fingers into their face.
yeah i pretty much agree with the OP. Are you familiar with special relativity as well as vector analysis and Newtonian physics? I believe to understand special relativity you have to understand vectors and also Newtonian physics.
Thanks for the post.
No I am not because those are not theses about what time is, but about 'behaviour'.
For instance, a theory - no matter how complex - about how someone behaves, is not a theory about what a person is.
Likewise, a theory about how things that are in time behave, is not a theory about what time is.
This kind of confusion - thinking that squarely philosophical questions are and have been answered by scientists - is, needless to say, rife.
I agree with the first four sentences. Physicists that I have known are interested in results, in the predictive power of their processes. I don't know how many, for example, worry about the metaphysics of the quantum world. Some do, of course, but when Feynman's integral gives the correct answer to a problem out to many decimal places that is considered success.
But time is far more than a philosophical question. We may never understand its fundamental nature. It may be beyond our limited powers of reason. But that doesn't argue against trying.
Rubbish.
Sort of, yes.
Consider the statement "All objects have a temperature at or above zero Kelvin". Interpreted from a realist's perspective, the sentence is synthetic and makes potentially falsifiable empirical claims, and in this sense is considered to be a "natural law".
But from the anti-realist's perspective, the statement is analytic and merely states that negative numbers play no role in our physical concept of temperature states; From this perspective, the statement is part of the convention of our physics language and in being fixed by convention isn't considered to be 'up for grabs'.
Yet as Quine pointed out, conventions often undergo dramatic revision every so often in order for a language to improve it's expression of new observations. But as Carnap pointed out, given any state of evidence, there is freedom as to the physics convention used. So there will invariably be disagreement as to whether any statement taken in isolation is false, true or meaningless. So in this sense there is indeed 'equivalence'.
Quoting Devans99
Unfortunately there isn't a single viable language of physics. Holistically, all viable languages account for the same observed phenomena, but each language suggests different prescriptions as to what are the most informative future experiments to conduct.
Thats actually not true. You can claim a very weak connection between any two entities or concepts and in some case a strong connection.
You might be right that there is almost no connection between special relativity and what time is but it is doubtful.
On a different note, i don't believe time travel would be possible unless there was someone who over sees what happened in the past. Time as a substance is a concept which basically says we observe objects and particles move. Since the past is forgotten as soon as it happens, someone would have to be responsible for recording it in exact detail in order to replicate it.
I believe time is understood and measured by the movement of particles. Since matter is limited by speed C (max speed C) and the summation of vectors probably plays a role (x + y + z cant exceed C) in why clocks tell time depending on how close to speed C they are traveling in a given direction, time is relative. This is one of many explanations of special relativity.
You sound very young. You have to understand people of many philosophies on this sight see science as the only way to answer any question on this site. Believe it or not various mathematical fields can be applied to any field of study including philosophy. Your favorite ice cream could probably be quantified through a systems analysis and design approach.
Without applying some field of mathematics or even a science, how do you expect to get a real answer other than "time is a banana split sundae."?
Folks, notice the slide Bartricks makes from "it doesn't exist", as used in the OP, to "it's not a thing", as used here.
Adopting for the sake of argument Bart's term "actual infinity", which strikes me as itself muddled...
(1) is wrong. A dimension need not be actually infinite, only potentially so.
(2)... well, we just do not know. But there is no obvious reason the universe could not be infinite.
This argument is philosophically more interesting.
Consider this argument, which purports to show that width is not a dimension...
1. If width is a dimension, then left and right are not intrinsic spacial properties.
2. Left and right are intrinsic spacial properties
3. therefore width is not a dimension.
Er no, that actually 'is' true. There's what a person 'is' and then there's what they get up to. Distinct theories.
Quoting christian2017
Arguments? Quoting christian2017
Question begging - you're just assuming that time is a stuff or dimension, that it is something we travel about in. I provided 3 arguments that appear to refute that idea. You've just blithely ignored them.
Quoting christian2017
Well that's nice for you. Once more: try actually engaging with the arguments. That is, try thinking rather than spouting.
And you sound very patronising.
Quoting christian2017
Yes, and that's a mistake. They're not doing the same thing - that's why 'science' exists as a distinct discipline.
These questions: what is morality? what is free will? what is truth? what is time? and so on, are 'not' questions science investigates. Why? Because you can't answer them by looking down a microscope. You have to apply your intelligence to them - that is, you have to reason. The questions have answers, but you're not going to find them by inspecting sensible matter ever more closely.
Quoting christian2017
And there it is: the arrogant dismissal of philosophy. You're one of those people you just mentioned above, aren't you?
Brilliant. I stand corrected.
Take a point and slide it. You get a line. One dimension. Only one number is needed to set out the relative position of two points - hence, length x
Take a line and slide it. You get a plane. Two dimensions. Two numbers are needed to set out the relative positions of two points - hence, the cartesian coordinates (x,y)
Take a plane and slide it. You get a volume. Three dimensions, measured with the coordinates (x, y, z)
Slide any of these temporally to add a fourth dimension.
Hence, time is a dimension.
Further, treating it as such is what enables the mathematics of mechanics. Mechanics posits that velocity is the change in position over time, hence quantifying time and so treating it as a scalar. This mathematical treatment is what Bart is rejecting. And along with it, the whole of physics.
Hm.
Bozos, notice that Banno can't read or quote accurately.
Oh, so dimensions are made of lines? I just drew some lines - I just created a dimension did I? Utter, utter tripe.
You don't grasp the basic point - nothing that is actually infinitely divisible can exist in reality. The 'idea' of it can exist - because ideas are not infinitely divisible - but it itself cannot.
So you sit in your cave writing bonkers definitions of 'dimension' on the wall to your heart's content - call a spoon a dimension if you want - the fact remains that any analysis of time that identifies it with something that can be infinitely divided is demonstrably confused.
Note too, I gave 3 arguments and one of them makes no mention of infinities.
Prove me wrong. Produce a reference, from Hilbert, that supports your point.
That would make me very happy. I would have learned something.
That argument is invalid. And you've missed the point.
I don't know if this is true, or not. But it is irrelevant. It's on a par with saying that there are never six things - always slightly more or slightly less; and then following it through by arguing that one can never buy a half-dozen eggs.
Sure. You do that. I'm making an omelet.
Do your own research. Prove me wrong.
And while you're trying to do that, try actually getting the point once in a while.
Actually fix it - present it again.
Nowhere in there does it make the claim to attribute to him.
Hence what you said was rubbish.
For a living, I assume? Tip: remove the shell, don't just mince it in with the egg.
Where? Nothing isn't evidence.
A dimension could be finite also - the dimensions on a fixed solid like a rectangle or torus are finite - and the universe may have a definite shape (in 4d spacetime). Dimensions are finite for example when you consider the surface of the earth as a 'universe'.
Our universe started expanding 14 billion years ago, suggesting that time and space dimensions are finite - there could be simply nothing beyond these boundaries - no time and space - so no measurement beyond these boundaries is possible/valid.
The reality you are aware of, perhaps. Is there nothing else?
If space was continuous, that would lead to a light year of space having the same informational content as a millimetre of space. That's absurd, hence space is discrete.
You don't seem to be getting the point. I am not saying that there is no more to reality than what I am aware of. I am saying that no actual infinities exist - that a hotel with an infinity of rooms is an impossibility. And thus, as time most certainly exists, it cannot be a dimension or stuff.
It's a simple and devastating argument. And there's the additional one - also devastating - about the intrinsic temporal properties of past, present and future.
Well, I can't argue with that. Above my pay grade. :brow:
If you say so. :roll:
So, once more we arrive at the conclusion that time is not a stuff.
Although presumably you'll say that it is a discrete stuff.
But then I want to know how that can be. To use space as the analogy: you have said that space is discrete. But how? Can you conceive of a discrete unit of space? Why can't it be halved?
No, I didn't just 'say so' - I've made arguments. You're not engaging with them - you're just expressing vague sceptical concerns about the whole project of using reason to investigate reality.
If we consider the particles within space, then it has a position - which can be regarded as information. A particle in a continuum has infinite decimal places in its position - infinite information. The same kind of infinity for a light year as a millimetre of space - hence the absurdity - the larger volume should contain more information rather than equal information - this is only possible if space is discrete. So I suppose this is an argument similar to Galileo's paradox.
The same argument for space applies to time (not unexpected given spacetime) - a year should contain more information than a second.
I would also point out:
- Matter/energy turned out to be discrete; why should we expect space to be any different?
- The only satisfactory answer to Zeno's paradoxes is discrete spacetime
Take this:
Quoting Devans99
Information is 'about' things, it does not 'constitute' the thing it is about. So, let's imagine a particle - a thing - in space. Now the particle is divisible. The space it occupies is divisible.
So, a particle - conceived of as an object extended in space - is infinitely divisible. Space, conceived of as a dimension, is infinitely divisible. And that's sufficient to demonstrate that our ideas of them are wrong, for no such things can exist.
Likewise with time.
The traditional way of thinking about time is just wrong, then. Anyone who thinks otherwise needs to address the refutations I gave, not simply assume the refutations fail and continue to persist with the traditional way of thinking.
What is a particle in a continuum? So .25 =.2500000... Only the first two decimal places have information worth having. I assume you are thinking of a particle on a real number line. If the particle lies on a line having only rational measures one can still get infinitesimally close since the rationals are dense in the reals. If the line has only integer measures, however, you do lose a lot of information. I guess this is what you are talking about. Have you studied math?
Actually i agree with you on that point. The other stuff you said really isn't even worth arguing over. If you don't see how mathematic principles can be applied to everything or almost everything, maybe i'll teach you some other time or maybe someone else will.
You won't have to look hard to find a pompous person on a philosophy forum. I believe to divorce systems analysis and design from any field of study is intellectual suicide.
I believe logic is a form of math and that phrases can in fact be systematically quantified. I'm not alone on this.
Once again you seem fairly young. Perhaps in 10 years we can have an intelligent conversation. Welcome to the online forum.
I find it hard to accept that, whilst I sit here typing, my fingers are passing through an actual infinity of positions.
If one rejects infinity in the large (? - infinite time and space), one must reject infinity in the small (1/? - the continuum) also.
Time is 'stuff' because:
- The physical laws of the universe are time-aware, so time must be something (IE 'stuff')
- Time has a start, so when time started something physical about the universe changed, so time must be 'stuff'
Given that spacetime is stuff, then it needs to behave like stuff and there is no stuff we are aware of that is infinitely divisible.
Spacetime also looks like a creation (see the BB) and it is not possible to create anything infinitely large or infinitely small.
But that's a tiny minority of possible numbers. The vast majority of numbers have infinite decimal places - that infinity of decimal places (=information) would be the same for the particle in a millimetre of space as for a particle in a light year of space which seems absurd to me.
Were you aware there is a one-to-one correspondence between any interval on the real line, no matter how small, and the entire infinite real line? It's easy to construct. You could do it quickly with a pencil and paper. :cool:
Maybe we haven't tried hard enough? I've seen arguments like this many times, projecting past failures to future attempts.
Thanks for the link. When I said that people don't find it difficult to conceive space as infinite I mean that it doesn't lead to an infinite regress like if time was infinite.
And I look forward to the day. Spacefoam anyone? :nerd:
Yes, they're not. Your fingers aren't what you think they are - if they were what you think they are, that is, objects extended in space, then they would have to pass through an actual infinity of postions in order to move. So they're not objects extended in space. Quoting Devans99
Well, time exists, but not everything that exists is a substance. Time is not stuff - not a substance - for the reasons outlined, namely that if it were a stuff there would be no intrinsic difference between future, present and past and because if it was a stuff it would have to extend infinitely - that is, actually extend intinity - which is manifestly impossible. (not all substances are extended, but time would have to be an extended substance - so it is extended substances, not substances per se, that contain actual infinities).
So, it seems to me that you are fallaciously inferring from the reality of time, the substance of time. But time can be real yet not be a substance - which is what we must conclude if the argument's I've presented go through.
Because scientists are not investigating what time is - they are just measuring stuff - whatever they say will be consistent with what time is.
It's true that there's at least one additional reason to think that time is not a substance (a reason to do with the intrinsic difference between past, present and future), but when it comes to the problem of actual infinities, the problem is the same. Space and time go the same way.
Time, if it is a substance, would have to extend infinitely because otherwise it would not be possible for an event to become ever more past for infinity. And that's manifestly absurd - no substance can extend infinitely.
But exactly the same is true of space as well. Space has to extend infinitely - how could it have a boundary? Whatever is outside the boundary would also be space.
And any region of space is going to be infinitely divisible.
One can just insist that this is not so - that is, one could, as Devans99 seems to be doing - reason that as no actual infinities can exist (correct), space must be reducible to discrete portions or atoms of space. But the problem with that is that it doesn't recognise that the problem is with space per se - any portion of space is going, by its very nature, to be divisible. I mean, try and imagine a portion of space that isn't divisible - it's impossible.
What we must conclude, on pain of simply refusing to face up to what reason is telling us, is that we are thinking about space and time incorrectly.
My bad. Sorry. I wasn't clear enough. Infinite space does lead to an infinite regress but that isn't a problem. People don't usually introduce infinite regress as an even a minor issue with infinite space.
However, infinite regress is a problem with infinite time because to get to this point in time we would have had to pass through an infinite past which seems inconceivable, infinity defined as it is.
The problem with infinite regresses is the 'infinite' bit. So, that we recognise an infinite regress to be a problem just underlines that actual infinities are problems - for an infinite regress just is an actual infinity.
Indeed.
No. An infinite regress, as I understand it, refers to the specific problem of an infinite task being impossible to complete. Infinity is the condition of being boundless.
You need to show me how space/time being infinite leads to an infinite task that can't be completed and that would be a problem.
For example, consider the first cause argument. Anything that has come into being needs a cause of its being. Positing another being that has come into being as the cause of those beings that have come into being starts one on an infinite regress. Why is that a problem? Why can't it be 'turtles all the way down'? Because you can't have an actual infinity of anything, be that causes, objects, actions.
How many natural numbers are there? Infinite yes? Is that a problem? No. Why? Because it doesn't lead to an infinite task.
How many points are there on a line? Infinite yes? Is that a problem? Yes. Why? As Zeno showed Achilles can't catch up with tortoise. An infinite task.
I can't even comprehend the terms that are being used.
Like here.
It isn't really a paradox anymore. Consider a line segment of length 1. It can be cut into length 1/2,1/4,1/8,1/16 and so on if we add up all the lengths, we get a line segment of length 1. A line segment is made up of countably infinite number of points. That's the way the real numbers work. That's also the reason why we don't have a smallest number "a" that is greater than say another number "b" .
That had me laughing out loud. :lol:
Now, if time were a cheese, it would necessarily be one sort of cheese, or it would be another. But it is not a cheddar. Nor is it a soft cheese. So, what sort of cheese is it? Feta? Too much salt.
Well l will see what l can say. Let's try this.
Er, no. It is the impossibility of an actual infinity that makes an infinite task impossible!!
Numbers aren't things. There aren't an actual infinity of numbers, rather they constitute a potential infinity.
Yes, that argument is valid and sound. Like mine.
I mean, I assume you accept that cheese is not a dimension? And I assume you acknowledge that the argument is valid? Yes?
Right. Now substitute the word 'cheese' for 'time'. Do any of the premises suddenly become false? No. Does the argument somehow become invalid? No.
So, what. Exactly. Is. Your. Problem?
What, exactly, are you having trouble with?
Yes, but there has to be a practical implication, an infinite task, that creates the difficulty.
In fact all paradoxes of infinity boils down to showing the practical impossibility of infinity.
I see no such problems in infinite space. What other alternative do we have if space is not infinite? Finite space, right? And the next question would be what lies beyond space? In fact infinite regress seems to be in favor of space being infinite rather than finite.
Given that time is just a spatial dimension we have limited access to, there should be no problem in imagining time too to be infinite.
No, even if there are no agents, you cannot have an infinite amount of anything. You don't make 'infinity' a problem just by adding 'task' to it.
Again, the reason infinite tasks are impossible is not because tasks are impossible, but because infinities are.
And yet we differentiate.
That's obviously question begging. You can't have actual infinities, so time is 'not' a dimension.
The same applies to space. You don't solve one problem by showing how it arises for other things.
Because we can't have actual infinities of anything, we need to rethink time and space - we 'must' be thinking about them in the wrong way. I am focussing here on time. Bringing space in - given that it raises many of the same problems - is unhelpful.
Time - time - is not a stuff, not a dimension. Why? Because thinking of it that way means it would instantiate actual infinities. That's sufficient to establish that it is not a stuff, not a dimension. But additionally, there would be no intrinsic difference between future, past and present (yet clearly these are radically different).
Limits, in mathematics, are calculations - tasks - that are infinite; they involve infinite steps.
You asserted that any task with infinite steps is impossible:
Quoting Bartricks
Limits show that your claim, again, is wrong.
Actually they don't. You are very wrong. But that's OK, carry on. :scream:
Tasks involving infinite steps are, indeed, impossible. That doesn't mean tasks are impossible. It means tasks involving infinite steps are.
I meant the real number line but the set of real numbers is uncountably infinite so l think l did mess up there. You can clear things up . :smile:
I hope it is correct now.
So, it's impossible. Says you.
Ok. Let's study this problem together.
Your claim: Time can't be infinite because of infinite regress.
Your reason: If time is infinite than we have an infinite past which raises the question "how did we reach this point in time?" Infinite regress.
Is this your argument?
No, that's not my claim. Look, I laid the arguments out.
Why is an 'infinite regress' a problem? Because.....you can't have actual infinities.
So, the claim is that you can't have actual infinities.
That's why infinite tasks are impossible. They're impossible because they would involve an actual infinity of something, namely tasks. Thus, as there can't be actual infinities, infinite tasks are impossible.
You seem hell bent on focussing on the wrong thing. So, it is not the 'taskiness' of infinite tasks that makes them impossible, but their infinite nature. And it is not the 'regress' of an 'infinite regress' that is the problem, but the 'infinite' bit. Why? Because - wait for it - you can't have actual infinities of anything.
Quoting TheMadFool
Again, that's not anything I've said.
An event - P - that is in the past is going to get more and more and more past, yes? Potentially infinitely, yes?
That, in itself, is not a problem. Potential infinities aren't actual infinities. So the fact that past events become ever more infinite - and will go on doing so forever - isn't, in itself, the problem.
Here's the problem. If time is a dimension, then in order to be able to accommodate the above, it would need to extend for an actual infinity. And nothing can be like that. So it isn't a stuff.
So, again, events in the past are becoming more and more and more past, potentially for infinity. But for that to be possible, 'actual' past - the time gloop these events are floating about in - would need to extend infinitely. And that's impossible.
Take rage. Some people are enraging. And the more you interact with some people, the angrier they make you. Is there any upper limit to how much anger one can feel? No. Anger can just go on getting more and more intense. There is no limit - it represents, then, a potential infinity.
Counting is like this. You can keep counting - 1,2,3 - on and on and on, potentially forever. There is no biggest number. Tell you what, try it - go and sit in the corner and start counting and tell me when you get to the biggest number possible, then report back.
Nuh. That 'd be too hard. Cooks have to work long hours at odd times. Not for me.
Because of...
Infinite tasks or the correct terminology being supertasks.
As you already know, super tasks, to be effective paradoxes and thus become problematic, are usually introduced with or within (a) limit(s) and are about infinitesimals which eventually spiral into infinity. For instance Zeno's paradoxes are about how a line is divisible into infinitesimals which lead to an infinity of points that must be traversed or considered.
The only way time as infinite is paradoxical is because of the supertask involved in reaching the present from an infinite past.
Any open line segment is in one-to-one correspondence with the entire real line, thus the points on it are uncountable.
Again, you change the topic.
Reiterating, you said:
Quoting Bartricks
SO, here is an infinite task: (1+½+?+...). The harmonic series. It diverges to infinity.
An infinite task, done.
More fun might be a convergent series - say 1-½+?-¼+... which adds to ln 2.
An infinity of tasks, done.
How are you not in error here?
cool. I agree a circle for instance doesn't exist in nature. Its a philosophical concept. There are things that are very close to being circles but nothing actually fits the definition described in a geometry text book.
That's my principle of engagement unless l get called out on my BS and the BS smells really bad and is clear as the day. Sometimes I just say
Go ahead punk ,make my day
Then blow off all the steam and bury everything into the ground.
We can actually, with help of computer software then print it out. Dud dahh !!
His claim is that an actual infinity is impossible. One way to make sense of that would be the problem of completing supertasks. That's all.
"..." doesn't constitute doing it. And saying "done" doesn't necessarily mean that it is done. Clearly, "..." symbolizes what is not done, not what has been done. The meaning of the ellipsis symbol is "unfinished". So your claim of"'done" is false.
Please explain what my fingers are if they are not 'objects extended in space'...
Quoting Bartricks
That's not true, under the moving spotlight theory of time, 'now' is a cursor that moves down a line (or around a circle maybe), so time can be a substance and we can still differentiate between past, present and future.
Quoting Bartricks
To divide something, you have to insert a piece of matter in-between the two parts. If space is made up of some sort of discrete mesh/grid, then it would be impossible to divide a mesh/grid node into two - the particle of matter exactly occupies one node of the mesh/grid at any time.
Quoting TheMadFool
There are an unlimited number of natural numbers - they go on forever in our minds - which is different from infinity - no matter how many times you add 1, you never get to a number called infinity. Only in our minds is it possible for something to 'go on forever' - if this occurred in reality, it would be akin to magic.
A point has length 0, say the line segment is length 1, then the number of points on it is 1/0=UNDEFINED. It is not infinite or unbounded, it is just UNDEFINED. It's not surprising considering a point is defined to have length 0 - so cannot exist - something with all dimensions set to zero clearly does not exist - so the question can be rephrased as 'how many non-existent things can you fit on a line segment' - an answer of UNDEFINED is exactly what you'd expect.
This tells us that any ‘point-like’ particle that exists in reality must in fact have a non-zero extension in space. So any real life line segment made up of real life points must have a finite number of points on it.
Quoting TheMadFool
The BB suggests that space maybe finite - space has been expanding at a finite rate for a finite time since the BB - so that suggests finite space (finite spacetime too). What lies beyond is pure nothing - there is no space and no time beyond the boundaries so nothing can exist.
Quoting TheMadFool
But nothing can exist forever in time, so it must have a start. See for example the argument I gave in this OP:
https://thephilosophyforum.com/discussion/6218/the-universe-cannot-have-existed-forever/p1
Quoting Bartricks
If time was finite in extent and discrete, then it would be a dimension without any actual infinities. Same applies for space.
Quoting Banno
Limits involve imagining an infinite number of steps which is distinct from actually performing infinite steps - actual infinity is unconstructable. See for example Thompson's Lamp paradox for the sort of nonsense results we get when performing the limit procedure out to actual infinity.
{ ..., 5BC, 4BC, 3BC, 2BC, 1BC, 0AD }
If time is infinite, how long does this sequence extend out to the left? Well it must extend out to encompass every number. But that's plainly an impossibility - at each point in the past (10BC, 100BC, a million BC, a trillion BC) you are 0% of the way to iteration of all the numbers (because any finite number divided by infinity is zero) - so no progress can ever made towards counting 'all the numbers'.
A believe in infinite past time is therefore akin to a belief it is possible to count 'all the numbers'.
Another Factoid: For those of you interested in the real line, did you know that if you have a cube, one foot on a side, say, there are exactly the same number of points within and on the cube as there are along one edge?
And if the Axiom of Infinity disturbs you, you would be frantic if you realized the consequences of the Axiom of Choice: " Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if the collection is infinite"
Doesn't that sound harmless? :nerd:
I am suspicious that arguments which have the outward form "time is thus-and-so, hence there must be a god" are actually of the form "There must be a god, hence time is thus-and-so".
Quoting Gregory Why? A new thread?
Interesting.
Quoting John Gill
That doesn't surprise me. Quoting John Gill Looks good. the fuss has settled down.
Well that would be a paradox, unless we return to the mathematical definition of a point as something with zero extents - it is logically impossible for such a think to have any existence. In each case (line, area, volume of the cube), I would contend that number of points is UNDEFINED. So it does not tell us much about reality except that mathematics (in this instance) does not reflect it.
With the definition of a point as something with non-zero extents, it all makes more sense: there would be less points on the line than the area, and less points in the area than in the volume.
Quoting John Gill
This is somewhat above my pay grade :grin:. I stopped reading up on maths after encountering the the axiom of infinity. It does, however, strike me that if there is an infinite number of bins, then the selection process never ends - it is not possible to complete the selection process - because the selection process requires infinite time - so therefore to my layman's mind, the axiom seems false.
That is, on a failure to teach mathematics well.
Hello, Meta. Sigh.
The harmonic series diverges (very slowly) to infinity. What is it that you think is not done?
But each time, it seems instead that a misunderstanding lies at the heart of each supposed problem.
It seems there are folk who read these and think "but you can't do that!", concluding that there must be an error somewhere. So they go on to philosophy forums and reinforce each other's errors.
Yes, Banno - write a book for us with the whole of the maths of infinity in it.
So my thinking is different from the traditional maths explanation. Please explain where I am going wrong?
See http://mathworld.wolfram.com/DivisionbyZero.html, especially the flippant answer from Derbyshire.
So mathematicians count points, instead of doing naughty divisions. Which has led to all sorts of interesting developments in dealing with infinities - especially uncountable ones.
The answer to your question, how many points are there on a line segment, is not that there are infinite, not that there are indefinitely many, but that there are uncountably many.
That was an excellent example. Thank you.
1. 1/0 = UNDEFINED
2. There is an infinite number of points on a line segment length one
That equates to a belief in both of:
1. 1/0 = UNDEFINED
2. 1/0 = ?
You see that obviously both [1] and [2] cannot hold at the same time... unless UNDEFINED = ? ... which is my belief.
"points" are an intriguing notion aren't they? Like "lines" with no thickness. There are theories of time that posit the non-existence of a "present point" But they are not useful in physics.
Yeah. No. I'm taking this example to my thread on critical thinking, because you are providing me with an excellent instantiation of the issue I was addressing there. .
https://thephilosophyforum.com/discussion/comment/362733
Are you a 100% positive or 99%? I like an opponent that is over confident. Look up the definition of a circle in a geometry book.
You are implying that:
- 1/0 is illegitimate
- 1/0 is uncountable / infinite
So which is it? It cannot be both.
No, I'm not.
You need to stop kicking people's asses. This is fun to watch though.
How so?
You state that 1/0 is illegitimate.
You also agree with the mathematical definition of a point as having zero extent.
Then the question of how many points there are on a line segment length one is perfectly valid:
- If its uncountable/infinite, then that suggests that 1/0 is legitimate
- If its undefined, then you are agreeing with me
Which do you choose?
No, it doesn't.
0 has no multiplicative inverse, hence 1/0 is an undefined or meaningless symbol. That's true.
That there are an infinite number of points in a unit line segment is true but ambiguous.
The cardinality of a set depends on the notion of bijection. But this is quite different from the measure of a set, which is a generalization of intuitive length. A measure (like standard Lebesgue measure) is a countably additive set function (among a few other requirements). Hence the measure of an uncountable union of points is not the (uncountable?) sum of the measures of the individual points. Indeed, this would lead to a contradiction, as you note, and assign a line segment of positive length (in the intuitive sense and in terms of the measure) a measure of 0.
Banno is frustrated because, well, mathematicians know their business. Any of us can give one of them hell about their philosophical interpretation of their professional discourse (if they bother to have one), but it's highly unlikely that a non-expert will catch them in a genuine contradiction. Mathematicians are experts when it comes to finding and using contradictions.
Math isn't philosophy. It's isn't metaphysics. 'It' can always retreat to formalism and utility.
It hasn't reached infinity, so it is not "done".
Yeah, it does - that's what the ellipsis is for.
As I said, ellipsis means unfinished. So using the ellipsis and claiming "it's done" is a false claim.
No. Here it means "keep going like this..."
https://en.wikipedia.org/wiki/Ellipsis#In_mathematical_notation
In other words you're never done.
OK, suppose you tell someone "keep going like this". At what point have they completed (are done) with that command? When they reach infinity?
Regardless of whether or not they understand the process, unless they "keep going like this" they cannot be said to have done the task. Understanding the task to be done, and doing it are two distinct things.
And, if someone thinks that understanding the task constitutes doing the task, as you apparently do, then that person actually misunderstands.
The task is to sum the sequence. The answer is that the sequence diverges to infinity. Hence the task is completed.
Your answer is not a sum. The task has not been done.
You were expecting what? An integer? What is it you see as missing from the answer?
The sum! Wasn't that the task, to sum the sequence?
1. Infinity is not a number.
2. Being undefined means we cannot assign any value to something.
3. Lim x->0 ( 1/x) is undefined
4. Lim x-> inf ( 1/x) is zero.
I think 3,4 will clarify all that confusion going on here.
That's a much better critique.
Quoting Metaphysician Undercover
The sum diverges to infinity. What more is there to say?
But "the sum diverges to infinity" is not "it can't be done"!
I'm 101 percent confident
Take the other example - 1-½+?-¼...
It converges to two.
You disagree?
Well, it's not a sum. And to say that the sum "diverges to infinity" says I can't give you that sum. So if you're not saying "it can't be done" then why can't you give me the sum?
Quoting Banno
I'd need a definition of "converges" before I'd agree to what you're saying, but if you mean comes closer and closer to, as you continue on the unfinished process signified by "...", then I'd probably agree. But this implies that it's not ever done.
Why not?
Wow. This goes on forever, doesn't it?
[math]\underset{x\to {{0}^{+}}}{\mathop{Lim}}\,\tfrac{1}{x}\text{ }=\text{ }+\infty [/math]
1/0 undefined. Suggest you move on to another topic. "infinite sum" is OK amongst professional mathematicians.
This passage is directly from a book of wittgenstein
N(0) is used for the cardinality of the set of natural numbers. Wittgenstein shows that the technique we learn in writing the usual whole numbers and writing N(0) are different and he concludes that we cannot say we have written N(0) numeral.
The ellipses aren't necessary. We have an increasing sequence of partial sums that is not bounded above. This sequence has no limit in the real numbers. 'Diverges to infinity' indicates more than merely a failure to converge by specifying that the sequence of partial sums is increasing. This is basic real analysis.
Consider also that proofs are finite objects. These finite objects and the things we do with them are inspired by intuitions of the so-called 'infinite.' And mathematicians aren't allergic to intuitive ways of talking. But in the end we have finite proofs that use a finite number of symbols. Such proofs can be (tediously) translated into dead symbols (bits if you like) and checked mechanically (by a computer, for instance).
This makes mathematics a prototypically 'normal' discourse, and perhaps explains the mixed feelings that metaphysicians have toward it. As I see it, the old dream of metaphysics is to do 'spiritual math' about matters of ultimate concern. Proofs of god, etc. But non-mathematical language seems caught up in time to a much greater degree. 'History is a nightmare from which I'm trying to awake.'
That only holds if one is a realist about the past who believes that the object of history is an unknowable reality in being unobservable and transcending present and future information.
In contrast, according to anti-realism the very truth of a past-contingent proposition reduces to present information. if the present state of information is ambiguous with respect to two historical possibilities, then according to anti-realism there is no matter of fact as to which historical possibility is true. Moreover, since the anti-realist never wants to claim 'perfect' knowledge of the past, he must insist that the past is infinitely extensible in a literal sense as and when new information becomes available.
For example, suppose Archaeologist A at time T unearths evidence E implying the existence of a fact F at time T', where T' < T. In stark contrast to the realist, the anti-realist considers A and E to constitute part of the very meaning of F, such that the truth of F is a function of T.
This is the category difference which makes Banno's claim of "done" false.
Quoting softwhere
Continuing with the Wittgensteinian perspective, the finitude of the proof would be dependent on the definitions of the terms. The definitions create the boundaries of meaning, required for the proof. If there is any vagueness, or undefined terms in the proof, then the proof cannot be considered as a finite object. Therefore it is very unlikely that we actually have any truly finite proofs, because definitions are produced with words, which themselves need to be defined, etc., ad infinitum. Vagueness cannot be removed to the extent required for the production of a finite object.
Quoting softwhere
Yes, we can class mathematics as "normal discourse", but to characterize "normal discourse", as working with finite objects of meaning, is what Wittgenstein demonstrates as wrong. This is why we must work to purge the axioms of mathematics from the scourge of Platonism, To consider proofs as finite objects is a false premise.
Bijection/one-to-one correspondence is a procedure that produces paradoxes like Galileo's Paradox, or the cardinality of the naturals is the same as the cardinality of the rationals. It is therefore to my mind an unsound procedure. Cantor did nothing to help our understanding of infinity IMO; he has lead us down the wrong path entirely.
Quoting softwhere
My (and Galileo's) point exactly - you fundamentally cannot measure something that is
uncountable/infinite - you would never finish measuring it - it is impossible to measure and claiming that bijection can provide a sound measure is ignoring the evidence (of paradoxes).
Quoting John Gill
The infinite sum concept in maths has definite problems, see here for an example:
https://en.wikipedia.org/wiki/Thomson%27s_lamp
To me the past is a deducible concept without referencing external realities - I have thoughts, these thoughts from a causal chain. The present exists, there are thoughts that I am no longer having, so the past exists. There are thoughts that I will be having so the future exists. I can label each thought with an integer. Assuming a past eternity, then the number of thoughts would be equal to the highest number. But there is no highest number, so a past eternity is impossible?
:smile:
I think if we all started reading Wittgenstein's Lectures on the foundation of mathematics, a lot of issues that come up here can be addressed in a good manner. As you have pointed at rightly. Wittgenstein regarded mathematics as a human invention, a finite calculus at most. But due to platonism, we sometimes give answers that answer different questions.
Wittgenstein further on even challenges the proof by induction used in mathematics.
That's simply an indication that we can do logic without knowing how the logic works. To know how logic works is a completely different issue. This question is Socrates' claim to fame. The artists and skilled craftsmen would claim to "know", because they had a technique which produced the desired results. This attitude extended into all fields, science, mathematics, even ethics and sophistry. Socrates demonstrated that these people who know how to do something do not know how it is that their activity brings about the desired end. Therefore their own claim to "knowing-how" is not grounded in anything, the activity is just a habit, and so is not real knowledge at all..
Say, [math]\sum_{n \in \mathbb{N}} f(n)[/math] is not a process like going shopping and returning home, it's a mathematical expression.
Convergence and divergence has concise technical definitions using the likes of [math]\forall[/math] and [math]\exists[/math].
I challenge you find and understand them. ;) At this point you might be in a position to launch critique.
By the way, you should know that this stuff has practical applications used every day by engineers, physicists and others.
Quoting John Gill
... when people aren't even trying. (Wait, I see what you did there.) :)
I suppose we might show the definition of [math]\lim[/math], and that it doesn't rely on [math]\infty[/math] other than implicitly by way of the neverending numbers.
Probably won't matter to the deniers, though, I sort of doubt it'd be worthwhile.
A process is a process. If your intent is to create ambiguity in the definition of "process", such that it is possible to have a completed process, which by definition has no end, then be my guest. Do not expect me to follow along with such contradiction though.
And if you back up, justify, such contradiction with the report that it has practical applications, I would reply that such applications are nothing more than sophistry, deception.
Disambiguation is what mentioned definitions do.
To me this frames infinity as an object that already exists, already has a nature. Philosophers can compare their intuitions in natural language, but mathematicians have got to make some rules.
So perhaps the burden is on Cantor's critics to offer a mathematical substitute.
Quoting Devans99
You misunderstand me. The measure of a set is different than its cadinality. https://en.wikipedia.org/wiki/Lebesgue_measure
Some of the confusion about Cantor seems to involve not realizing that mathematicians also have other ways of comparing sets. Measure is more intuitive than cardinality. [0,2] has twice the measure of [0,1] and yet the same cardinality. And then also have homeomorphisms in topology. These concepts each treat of a different quality that sets can have in common. With people we can talk about height, eye color, shoe size, etc. It's the same with sets.
Quoting Metaphysician Undercover
I find it hard to make sense of this. Proofs are obviously finite objects, or we could never finish reading or writing one.
Wittgenstein can't demonstrate that this or that as wrong mathematically. He comments ultimately about interpretations of the calculus (game of symbols). I like some of his critiques. And I also like intuitionism, finitism, constructivism. The philosophy of mathematics is deep and complicated. A person can have philosophical doubts about mainstream mathematics and still be good at it.
And then most people never learn pure math. They learn algebra, trig, calculus, applied linear algebra. This stuff is fairly intuitive and incredibly useful. To me anti-Cantorian passion suggests a love for pure math in that it wants to get the infinite 'right.' The door is always open. A person could construct a system. To be math, it would need rules. This guy actually tried to deliver a replacement for the foundations he objected to: https://en.wikipedia.org/wiki/L._E._J._Brouwer
A few people work on systems like that. But most people who use math don't care at all about philosophical disputes. Math is just a tool that they use according to conventions.
It can and has all been done with symbols. It's like a game of chess. In practice words are used to abbreviate formal proofs and aid the intuition.
[quote=link]
A formal proof is a finite sequence of formulas, each member of which is either an axiom or the result of applying a rule of inference to previous members of the sequence. Typical rules of inference are modus ponens and the substitution of equals for equals. A grammar for formulas, a collection of axioms, and a collection of rules of inference together define a logical theory.
For the usual theories of mathematics, e.g. set theory or number theory, it is a relatively modest exercise to write a program called a proof checker that will check, in a reasonable amount of time, whether a given sequence of formulas is a proof.
[/quote]
http://www.cs.utexas.edu/users/boyer/ftp/ics-reports/cmp35.pdf
This isn't surprising. Unless proofs could be verified, math wouldn't be a normal discourse. It would just be a quasi-literary metaphysics with mathematical themes.
But as I understand it, maths frames infinity as an object that already exists (axiom of infinity). I believe that axioms should be more than assumptions - they should be self-evident truths - and what is self-evident about the existence of an actually infinite set? Parallel lines not meeting I can swallow, but an actually infinite set?
Quoting softwhere
I am not a mathematician but I would imagine that the vital areas of mathematics (IE calculus) could function well enough using just the concept of potential infinity. I do not see why the concept of actual infinity is needed - it just leads to paradoxes.
Quoting softwhere
I am not disputing it is possible to measure intervals, I am disputing the common mathematical claim that there is an actually infinity of points on a line segment length 1.
How many points do you claim there are on a line segment length 1? The answer must logically be one of the following:
1. Infinite number
2. Finite number
3. Undefined
(there are no other possibilities)
If it is [1], that means 1/0=? which is nonsense
If it is [2], then a point must have non-zero length which is not the definition used in maths.
So I contend it must be [3].
I think that's a good question. To me it's fair indeed to operate at this level and engage in a philosophical debate about the rules of the game that human beings freely decide upon. I'm not a specialist in set theory, but certainly a set theorist wants to construct more familiar mathematical objects like natural numbers.
[quote=Wiki]
the other axioms are insufficient to prove the existence of the set of all natural numbers.
[/quote]
Basically they had to have it if they wanted the natural numbers, and they had to have the natural numbers. But others have wanted to take the natural numbers as fundamental.
[quote=Erret Bishop]
The primary concern of mathematics is number, and this means the positive integers. . . . In the words of Kronecker, the positive integers were created by God. Kronecker would have expressed it even better if he had said that the positive integers were created by God for the benefit of man (and other finite beings). Mathematics belongs to man, not to God. We are not interested in properties of the positive integers that have no descriptive meaning for finite man. When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself.
[/quote]
https://en.wikipedia.org/wiki/Errett_Bishop
'God created the positive integers' is just a metaphor for their obviousness to intuition. Other comments involve constructive/intuitionistic logic. There's a fascinating ideological purity in this that appeals to me. But the mainstream chose otherwise. At the same time, most mathematicians don't worry themselves about this stuff in my experience. (I stand out by questioning the meaning and value of the game, but then I should have studied philosophy instead.) They learned a certain set of rules that they are happy with. The rest is disreputable 'philosophy,' inferior because it's just 'opinions.'
Quoting Devans99
I still think you are confusing mathematical and metaphysical claims. What do you mean by 'actually infinite'? Math isn't done with ambiguous philosophical terms. You can complain that the axiom of infinity violates your intuition (fair enough), but it's trivial to show that [0,1] is an infinite set using 'the rules.' What that means philosophically or metaphysically is another question entirely. It is essential for math that it function independently of metaphysics. Once the rules are chosen, there is no more room for confusion or ambiguity. It is a dead machine.
Quoting Devans99
What you neglect here is the ambiguity of 'infinite number.' This is pre-mathematical metaphysical ambiguity. In truth, I think you are guilty of the very think you accuse mathematicians of. If one wants to be strictly and even mechanically logical, then one needs strictly and even mechanically defined terms. This is precisely why notions of cardinality and measure were painstakingly developed within a 'realm of law.'
Cantor's most famous breakthrough was showing that one notion of infinity was not enough. His work has consequences for theoretical computer science. It's not only artistically charming. Whatever limitations or blemishes one finds in mainstream math, it's a spectacular structure.
Also, choice #1 does not imply that 1/0 = infinity. Saying so is pseudo-math. Since your attracted to this stuff, why not study some math? Even if you just want to criticize it, your criticism will only be plausible if you can project a basic competence --that you actually know what math is and how it operates from a position within math. Fascinating criticisms of math can be made, but all the criticisms I've seen online by the untrained have so far been just projected misunderstandings. I don't mean to be offensive. I like the critical philosophy of mathematics. It just has to know its target in order to hit it.
The problem I see is that (applied) mathematics forms the basis for our understanding of reality. So scientists pick up definitions and theories from maths and apply them to the physical sciences.
Now the set of natural number exists purely in our minds - my believe is there is nothing in reality akin to it. So there is this impossible concept which is taken from maths and is being applied in the physical sciences - producing erroneous results - cosmology is the biggest offender.
I feel a way of defining the natural numbers without resorting to questionable concepts like actual infinity would be the way to go.
Quoting softwhere
It does imply that 1/0 = ?, we need only pre-school maths to arrive at such a conclusion:
1. Maths claims an infinite/uncountable number of points on a line segment
2. A point is defined by maths to have length 0
3. The line segment in question is length 1
4. So we must divide the interval into 0 equal pieces to find the number of points in it
5. Hence the number of points is 1/0 = ?
Where is your dispute with the above reasoning?
The problem is that you think you can do philosophy of math with only pre-school mathematics. If you don't know how to read and write proofs, then you don't really know what math is.
Quoting Devans99
I've already discussed that. Measures are countably additive set functions. Cardinality is actually important here. But measures will make no sense without a mastery of basic real analysis. Usually one learns this over a course of years. It's like learning to become fluent in a language. Since inexpensive Dover books are easily available, I won't go into great detail that's likely to be ignored anyway.
If you actually want to resolve your confusion rather than install it as a work of genius, you'll just have to learn some math.
It should please you to know that you are at the point mathematicians were at two and a half centuries ago as they pondered what infinite sums meant. Does the sum S=1 -1 +1 -1 + 1 ... make any sense? Indeed, the partial sums are 1,0,1,0,1,... like that pesky light switch (which is seen as pretty silly these days - not a "definite problem"). After Cauchy and Weierstrass and others formalized convergence criteria there was still the amusing question of series that oscillated like the one above. Then other mathematicians developed summation processes (SP) that had the following features: If a series converged in the normal sense, it must converge to the same value in the SP , but some series that did not converge in the normal sense might "converge" in this new way.
For instance, one such SP is to add the first n partial sums and divide by n. If the given series converges as n gets larger without bound, this new process will converge to the same value. But in the above conundrum, note that this process yields a limiting value of 1/2. And some mathematicians long ago stipulated that value for the series, before Cauchy and Weierstrass had their says.
As for measures, although the simplest is the length of an interval on the real axis, used in the Riemann integral, they get much, much more complicated and abstract and are used in what are called functional integrals. I suggest you don't go there. If you are curious, go to my page in researchgate and pull up the note on functional integrals.
This has bothered me since you first brought it up not a while ago. I'm not a mathematician but 1 here is a length and when you divide a length you don't get a point. What you get is another length.
Also, a point isn't defined in terms of how big/small it is i.e. it isn't dependent for its existence on its own dimensions which as you rightly pointed out is zero. A point is actually defined in terms of its distance from the origin (0,0) or some other reference point.
Dividing a length by a point doesn't make sense in the same way as dividing Tom, Dick, Harry and John by Dick or Harry doesn't make sense.
What lies beyond the boundary of "finite" space? Can an infinite space not expand?
Imagine three galaxies in infinite space A, B, and C. Suppose the distance between them is 4,000 lightyears. Can't the space between these galaxies increase, not because they're moving but because space is being created between them. In other words I see a possibility of an infinite and expanding space.
Quoting Devans99
What about time itself? Did it have a beginning? If space can be infinite and time is "just another" dimension, and if space can be infinite can't time be too?
The measure of the interval [0,1] is 1 and the measure of the interval [0,2] is 2. This way of classifying size also leads to the conclusion that a point must have non-zero length:
length of a interval = pointsize * pointnumber
Neither of 'pointsize' and 'pointnumber' can be zero because then the measure of the two intervals would be equal (zero in both cases). So a point cannot have zero length.
?
I still feel maths does not currently have a complete understanding of infinite series:
- At any intermediate point in the evaluation of Grandi's series, it always has a sum of 1 or 0. Therefore logically the final sum can only be 1 or 0 - there are no other possibilities
- But mathematical methods for evaluating series yield the sum of 1/2
- Maths calls the series divergent as the individual terms do not approach zero
- But if we knew whether ? is odd/even we could evaluate the series
- But my contention is there is no such thing as actual ?
- So IMO the final sum of a series (taken to ?) is a meaningless concept (in some instances)
Quoting TheMadFool
OK, so your interpretation is (as I understand it) that that a line segment is not composed of infinite points, but is composed of sub-lengths. I am in agreement. I would point out that the length of a sub-length cannot be zero else all line segments would have the same size.
Quoting TheMadFool
If the distance between them is currently 4000 ly and the universe is expanding. then there must have been a time in the past when the distance between them was 3000 ly, 2000 ly, 1000 ly, 0 ly. At the final point, when the galaxies are co-located, the universe cannot be expanding. So I think that infinite expansion is impossible. I believe there are some cosmologists who disagree with me on this.
Quoting TheMadFool
It comes to a question of origins. I believe that there must have been a first cause for everything in time (the cause of the BB probably). Then the obvious question is what caused the 'first cause'. We could answer that by introducing another cause to cause the 'first cause', but then we would need another cause, and another, so we end up with an infinite regress with no ultimate first cause of everything - which is impossible.
So there seems to be a need for a first cause and there cannot be an empty stretch of infinite time preceding the first cause - then there would be nothing but emptiness to cause the first cause - which is impossible.
So there seems to be a need for an 'uncaused first cause'. How do you get an 'uncaused cause'? Well causality is a feature of time, so placing the first cause beyond time seems to be the only way to have an 'uncaused cause' - then there is nothing logically or sequentially 'before' the first cause - the first cause has permanent uncaused existence. The first cause is then the cause of / creator of time (time must have a start).
I can sum up the argument with an altered version of the PSR:
- Everything in time must have a cause
Which leads naturally to a timeless first cause.
This also leads to, incidentally, an answer to 'why is there something rather than nothing?' - there has always been something - uncaused and beyond time - there is simply no 'why?' for something that is uncaused.
The issue here is that there is an incommensurability between distinct spatial dimensions. Pythagoras demonstrated that the ratio between two perpendicular sides of a square is irrational. The same type of irrationality arises from other two dimensional figures, like the circle, with the irrational pi.
This incommensurability is extremely evident in the relationship between the non-dimensional point, and the one dimensional line, according to TheMadFool's explanation.
Whenever we add another dimension to our spatial representations we add a new layer of complexity to this fundamental incommensurability, such that by the time we get to a four dimensional space-time the irrationality involved is extremely complex. What is indicated by this fact, is that our representation of spatial existence, in the form of distinct dimensions, is fundamentally flawed.
Likewise, a 1-dimensional line cannot be the constituent of a 2-dimensional plain - the line has length but zero width so it cannot be the 'parts' of a plain (which has non-zero length and non-zero width).
As you are both mathematicians, I hope you don’t mind if I take the opportunity to explain my concern a bit further. I feel there are some deep problems with infinity in the foundations of mathematics and it seems to me that many mathematicians are very invested in their discipline and their hard-won knowledge - so that they are not usually eager to confront such issues.
For example, take the extended complex numbers - the set of complex numbers plus ?. The definition used for ? is z/0=?. Now you can call that ‘an assumption’ if you like (and a pseudo-justification in terms of limits can be given) but it is plainly a wrong assumption. I believe there are then fields of maths (like complex analysis) which build on the idea of the extended complex numbers. Then people in the physical sciences build further theories based on these ideas. The net result is whole vertical slices of human ‘knowledge’ which are based on wrong assumptions and are therefore not valid knowledge.
Similar bad assumptions to the above example can be found in the hyperreal numbers and the projectively extended real line. Another example, already discussed above, is the axiom of infinity from set theory - the assumption of the existence of actually infinite sets of objects. It is a bad assumption to make and set theory is based on that bad assumption. Many things in maths and science are then built upon the foundation of set theory. Again we have whole swaths of knowledge based on bad assumptions - all that ‘knowledge’ is therefore not valid.
I am not a mathematician so I cannot call out more examples than this, but I’d be surprised if there are not more. It is therefore not surprising (I hope) that laymen such as myself are disinclined to try learning more of advanced mathematics - my (admittedly limited) experience of such is that (what can happen) is early on, in the foundations, a bad assumption is made and then a body of interesting, complex but ultimately invalid results are derived based on that bad assumption.
I feel mathematics has a responsibility to the rest of the physical sciences to keep the assumptions reasonable. By reasonable, I include assumptions that are provisional - they may lead to interesting, but provisional results (eg non-euclidean geometry). I also class as reasonable assumptions that widen the scope of traditional mathematics, such as the introduction of new types of numbers (eg complex numbers). But assumptions that are plain wrong/bad (counter logical) lead nowhere useful, lead other folks (in the physical sciences) astray, and result in lots of clever folk wasting huge amounts of time on wild goose chases (eg a good portion of modern cosmology is like this IMO).
Your thoughts?
What do you mean? The ratio "of"? Take a square with sides =1. The 1/1=1. Are you talking about the hypotenuse of a right triangle? Like each side = square root of two?
Quoting Devans99
This is wrong, and shows how difficult it is to debate with you. Apparently you know so little of math you cannot even frame your beliefs correctly.
https://math.stackexchange.com/questions/2424005/what-does-infinity-in-complex-analysis-even-mean/2424111
Software, what is your math background?
“Nonetheless, it is customary to define division on C ? {?} by
z/0 = ? and z/? =0”
https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers
"There are, however, contexts in which division by zero can be considered as defined. For example, division by zero z/0 for z in C^*!=0 in the extended complex plane C-* is defined to be a quantity known as complex infinity.”
http://mathworld.wolfram.com/DivisionbyZero.html
Here's a comment from the talk page on Wikipedia. Not mine.
[i]"The Riemann sphere is just the complex plane with an extra point added in, called the point at infinity. For analogy, look at the real line. There, when dealing with limits, it is convenient to pretend that there exist two points ? and -? which are endpoints of the real line. Then ?+?=?, and all other formal rules makes it easier to deal with limits without worrying much about particular cases of infinite limit.
In the same way, one can pretend that all rays in the complex plane originating from 0 actually have an endpoint, and they all eventually meet at infinity, a point far-far away (not accurate as Elroch mentions above, but helpful in imagining things).
The Riemann sphere is not the same as the usual sphere, but they are topologically equivalent. Imagine a normal sphere, remove the north pole, and make the obtained hole there larger and larger (assume the sphere is made of very flexible rubber). Eventually, that sphere without a point can be flattened in a plane, the complex plane. The original north pole corresponds to the point at infinity in the complex plane.
There is only one Riemann sphere, as the point at infinity is just a symbol, its actual nature is not relevant. "[/i]
You need to move away from your fixation on the symbol z/0, which you seem to believe has some quality that disrupts reasoning in mathematics. I suggest you look into the Axiom of Choice, about which there is genuine relevance to mathematical thought.
But you did score a point here! Congrats. :grin:
We are talking about the nature of time, whether it has a beginning or end specifically. Such a conversation is intimately linked to the existence or non-existence of Actual Infinity. Maths treatment of the subject could hardly be described as definitive - a set of non-sensical assumptions IMO. Notice I have highlighted the phases 'pretend', 'without worrying much'... such words hardly inspire confidence...
There is no largest number - numbers go on for ever in each direction - there is no such number as ? - so a mapping of non-existent points at actual infinity to the north pole of a sphere is a nonsense procedure.
And yes z/0=? appears to have 'valid' (!) applications in maths:
"The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=? well-behaved."
- https://en.wikipedia.org/wiki/Riemann_sphere
There are certain contexts where such (sort of "artificial") constructs are convenient, as long as that's understood.
I don't think you can simply lean on this sort of convenience here and call it a day.
'Foundations' is a misleading metaphor. In general, applied math comes first. Calculus was invented and successfully applied long before a careful definition of real number system was given. Pure math isn't, in my view, the genuine foundation of applied math. Instead human beings just trust tools that tend to give them what they want, reliably.
If you live in the city, consider the tall structures of concrete and steel. Why is it that they don't tumble down? Aren't they based on the axiom of infinity? No. Formal set theory is arguably more of an aesthetic enterprise. Engineers don't need it. Pure math is its own beast, and I suggest that its prestige is parasitic upon that of the technological enterprise.
Quoting Devans99
I think you'd have to make a case that physical scientists are being led astray, a claim I find less than plausible. I know very little about modern cosmology, but some of your other comments make me think that you are wrestling with Kantian themes. Is a beginning of time thinkable? No. Is beginningless time thinkable? Also no. I'm sympathetic to that kind of thing. I like instrumentalism as a way to make sense of science. To me neither math nor physical science offers 'metaphysical' truth (replaces philosophy).
You are asking math to do what it cannot do and does not claim to do, namely metaphysical philosophy. If you want a deep investigation of time, look into Hegel or Heidegger or Kant. The results of math being so certain comes at the cost of their significance being indeterminate.
[quote=Russel]
Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
[/quote]
I think this is an illuminating exaggeration. Intuition is important in math, but math is protected from the endless misunderstandings found on philosophy forums by rules and exact definitions.
I will humor you. The number of points in [0,1] is uncountably infinite. Measure, however, is only countably additive.
http://mathworld.wolfram.com/CountableAdditivity.html
That means you can't split [0,1] into points and then add the measure of the points to get the measure of the interval. You can, however, split [0,1] into [0, 1/2) and [1/2,1]. In this case all is well. One can even measure Q, because Q is countably infinite. And the measure of Q is 0. Note that the two most famous kinds of infinity (that of N and that of R) actually do come into play right away in measure theory.
Keep in mind that I might as well be talking about the rules of chess. You can say that these are all fictional entities. It is true that theoretically this is the foundation of statistics. But I suggest that human beings would still apply math without careful justifications that few of them ever study in the first place. We trust tools that work. Consider Hume's problem of induction. All of our technology is based on a gut-level faith in the uniformity of nature.
Yes.
Quoting Devans99
:chin:
At some point I think this leads us into the philosophy of language. How do the signs 'actual infinity' function in our community? Is there ever some sharp meaning in our head? Maybe the question isn't binary, one of clear existence or non-existence. For why should 'exist' be any less complicated semantically than 'actual infinite'? I
I understand where you are going. But how sensible is this time before time? I find it as questionable as intuitions of actual infinity. Personally I think human cognition runs aground on issues like this. It's as if we just weren't equipped for such questions. If time and space are automatic intuitions, then we run the risk of talking non-sense. At the same time, we can create mathematical models that defy intuition that nevertheless have practical power.
This is why I like instrumentalism as an interpretation of science. And I also think that reality as a whole is inexplicable on principle. Some principles always must remain 'true for no reason.' They are patterns that are just there. Later we may derive them from still larger patterns, but this just expands the circle whose outermost ring is contingently true.
I would contend that set theory gives an unwarranted legitimacy to actual infinity which influences the physical sciences. Some cosmologists are obsessed with actually infinite time and space and they lean on concepts borrowed from pure maths to justify such infinite models.
Quoting softwhere
Measure theory does not seems to provide any justification for the above claim - neither do I see any justification anywhere else in maths.
Quoting softwhere
There are a number of good arguments that all point to a start of time (one example: perpetual motion is impossible, things are currently in motion -> a start of time). If there is another type of 'time' 'before' our time, the same arguments apply to that 2nd type of time - it must also have a start. Obviously there cannot be an infinite regress of such times, therefore we seem to be left only with the possibility that something must exist that is timeless /atemporal.
You maybe right stating 'human cognition runs aground on issues like this'. Philosophers have grappled with the nature of timelessness for centuries and I have not yet encountered a satisfactory explanation of what it could be.
I imagine our universe as a 2d space time diagram - a plane of finite dimensions. Then I imagine a point off to the side of this plane that represents a timeless thing. Then I imagine there is a mapping from the point off to the side to each point in the plane - the timeless thing experiences everything in time in one 'eternal now' and can interact with anything in spacetime. Of course this does not really shed any light on the nature of the point - the timeless thing.
- A point has zero length according to maths definition
- But according to my intuition, a point must have non-zero length
Do you suppose there's a reason why points are zero-dimensional?
How would we define distance? The beginning/end of one point to the beginning/end of another point? Why not just consider the beginning/end as zero-dimensional points?
I find the concept of a dimensionless object difficult - it has no extents so it cannot have any existence - how can any sound reasoning performed with a non-existent object - assuming its existence (in order to reason with it) leads straight to a contradiction?
Quoting TheMadFool
Well the concept of 'measure', as alluded to above by @softwhere, seems to be math's answer. But measure theory does not seem (from my very limited knowledge of it - ?) to provide a justification for treating a point as dimensionless (or that there are infinite points on a line segment).
To define distance in a conventional sense is to have a unit of measure and a zero dimensional point is not a valid unit of measure - if we say a point has zero length and try to use it as a measure, then the measure of all line segments, no mater what length, is UNDEFINED.
I'm not even sure it is correct to say the beginning/end of a line are points - points don't exist - so would the line even have a beginning/end?
The important thing to grasp is that math doesn't ultimately work with mere intuitions of what a point 'really' is. Such intuitions can guide the construction of a formal system, but the math itself is a definite formal system. That's why serious finitists and contructivists offer new exact systems. If it isn't a system that's as a dead and machinelike and trustworthy as the rules of chess, then (roughly) it isn't math. Because then it isn't a 'normal discourse.'
This is why a person has to learn to read and write actual mathematical proofs to really know what math is. So I encourage you to get a book on proof writing and reading. It will be fascinating, and your criticism will have more relevance.
I did a cursory reading of Euclid's definition of a point: "that which has no parts" which I suspect alludes to points being zero dimensional.
The usual way points are expressed in Cartesian plane is by an ordered pair (a, b) which to me means points are intersections of lines, in this case the lines x = a and y = b.
Lines are not dimensionless, they have length, so shouldn't be a problem for you. After all you seem to have an issue with the dimensionless and lines have a 1 dimension viz. length. However, lines don't have width i.e. lines have one and only one dimension which is their length. This won't be a problem for you either since even though lines lack a width, lines have a dimension, length.
Now, consider the intersection of two non-parallel lines. They intersect on their widths and not their lengths. Such intersections being points can you now see how points can be zero-dimensional?
This expression sounds very anti-realist to my ears. Namely that the past is deducible i.e. in some sense a living construction out of present sense-data and current activities, as opposed to being an inductively inferred and immutable hidden reality that cannot be observed.
The conceptual distinction between the inductive inference of causes from effects versus the deductive 'construction' of effects given causes seems to lie at the heart of disagreements between the realist and anti-realist. The latter wants to bring these two concepts much closer together by interpreting causal induction constructively as a generalised form of deduction.
Quoting Devans99
Yes, the realist thinks of "past eternity" metaphorically in geometric terms, as an infinitely long line beginning at, say, zero and ending at positive infinity at the "the present", which obviously cannot be constructed. The anti-realist can reverse this metaphor by labelling the present with zero, and considering the past to be 'created on the fly' as and when evidence of the past becomes available.
And of course, you presumably mean that you have present thoughts which you interpret as being 'past-indicating' and present thoughts you interpret as being 'future-indicating'. Although recall that an appearance per-se does not refer to either the past or future, as exemplified by a randomly generated image that by coincidence looks historical. And recall that we can doubt the veracity of our memories if we judge our present circumstances to contradict them. So we cannot make an immediate identification of appearances, memories, thoughts or numbers with points on a physical-history timeline.
I class myself as a realist but a finite realist, which I consider to be a more 'materialistically real' proposition than a realist who believes in actual infinity - for example of past time:
- If time has a start then it was 00h:00m at the start of time and current time is given by elapsed % 24
- If time has no start then it is UNDEFINED and all current times are also UNDEFINED, so no time
Quoting sime
So you would hold that even the temporal succession of old thoughts followed by present thoughts followed by new thoughts could be an illusion?
Thoughts are constantly happening so that leads one to have an ever expanding history of thoughts. You hold that even that ever expanding history could be an illusion?
It is hard to argue against such a viewpoint. Denying the evidence of the senses as real is one level of anti-realism. Denying the veracity of our own thought is another level of anti-realism. The second seems impossible to counter logically?