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Tasalsul is the belief that the present moment is preceded by an infinite number of past events. In proving the existence of God, Imam Al-Ghazali offers three arguments for the impossibility of this belief in his Al-Iqtisad Fi Al-I’tiqad:
| All of the Philosophers agree that the bodies of the world are inseparable from events, and yet they claim that the world is beginningless. So if it is said: “you claim that whatever is inseparable from events, is itself emergent. What is your proof for this?” We say: if the world were beginningless, while being inseparable from events, then this would mean that an infinite number of past events occurred, entailing that the number of orbits each celestial body completed be infinite[1]. This is impossible, because whatever entails an impossibility is itself impossible, and we will now show that three impossibilities follow from this belief.
The first if this were true, then it would mean that an endlessness [i.e. the infinite number of past events] was “completed”, “concluded”, or “came to an end”, and the three expressions are synonymous. So it would follow that an endlessness came to an end. And this is clearly contradictory. The second if the number of orbits completed by each celestial body were not finite, then the number of orbits completed would have been either: even, odd, neither even nor odd, or both even and odd. Each of the four possibilities is impossible in the case of an infinite number of orbits, so the completion of an infinite number of orbits is impossible. It is impossible for an infinite number of completed orbits to be neither even nor odd, or both even and odd. An even number is one that is divisible by two, like ten for example. Whereas an odd number is one that is not-divisible by two, like nine. And any amount is composed of a multiplicity of singulars, such that this amount is either divisible by two, or not. Therefore, it is impossible for an amount to be neither divisible nor not-divisible by two, and it is impossible for an amount to be both divisible and not-divisible by two. It is impossible for an infinite number of completed orbits to be even, since an even number is so because it lacks one, such that if this one were added to it, it turns odd. But how can an infinite amount lack anything? It is also impossible for this amount to be odd, because an odd number becomes even by adding one to it, so it can only remain odd as long as it lacks this one. But then again, how can an infinite amount be lacking in anything? The Third it would also follow that there be two amounts, each infinite, while one is lesser than the other. And it is impossible for an infinite amount to be less than another. This is because the lesser amount lacks the difference that if added to it, would make it equal to the other. But how can an infinite amount lack anything? To make matters clearer: they [the Philosophers] claim that Saturn completes one orbit [around the Earth] every thirty years, while the Sun completes one orbit every year. So the number of orbits that Saturn completes annually, is a thirtieth of the Sun’s. But if one proposes that Saturn has been orbiting since eternity past, then the number of its orbits would have been infinite, even though this amount is also less than the orbits of the Sun. After all, a thirtieth of an amount is obviously less than the full amount. Similarly, the moon completes twelve orbits every year, which means that the Sun completes twelfth as many orbits as the moon. Each amount is said to be infinite, while each is less than the other, and this is a clear impossibility. |
– Al-Ghazali, Al-Iqtisad Fi Al-I’tiqad, page 24
[1] The Philosophers maintained that the celestial bodies have been orbiting the earth since eternity past. So the Imam argues that if the world were beginningless, then this would mean that the number of orbits each celestial body completed, would have been infinite.
Keys to the Unseen 
salam,
the three objections Ghazali raises are weak, based on misunderstandings of the positions of the hukama’.
what we mean when we say that the past is infinite is that:
for any past event you care to choose, there’s a event before it, without end.
this disarms the first objection; for all i mean when i say ‘an infinity of past events have ‘occurred’ is that ‘every past event (you want) is preceded by some other event’, ad infinitum’. nothing absurd about that.
as for the second objection, it fails because it treats what is infinite as an actual number.
as for the third objection, the claim there is just false – one infinite can be greater/lesser than another. example: the set of natural numbers (infinite) is less than set of real numbers (also infinite).
Wa ‘Alaykum Al-Salam,
“for any past event you care to choose, there’s an event before it, without end.”
Meaning: this chosen event is preceded by an infinite number of other events. Such that this infinite series of preceding events concluded, before this chosen event emerged into existence. And this is absurd because an infinite cannot be concluded, since it is endless by definition.
I don’t see how your rephrasing is an answer to the first objection.
“as for the second objection, it fails because it treats what is infinite as an actual number.”
If you accept that infinity is not a number, then you can be asked: new events are being added to the series of past events as time progresses, so if this series of past events were infinite, and infinity is not a number, how can addition be performed on a non-number?
“as for the third objection, the claim there is just false – one infinite can be greater/lesser than another. example: the set of natural numbers (infinite) is less than set of real numbers (also infinite).”
In of themselves, numbers are abstract objects that don’t exist in the real world. Therefore, when one claims that the set of natural numbers is infinite, the claim isn’t that there is an actual infinite number of existents. Rather, that whenever you mention a natural number to some person, this person can give you a natural number greater than the one you mentioned. However, no matter the number of naturals this person lists for you, he will never be able to finish listing an infinite number of them (for the reasons mentioned above).
salam,
if by ‘concluded’, you mean ‘passed’, elapsed’, etc., then all you’re saying is:
‘before this moment/event, an infinite number of moments/events’ passed or elapsed’
and, i said, this means:
‘for every in the past you care to choose, there’s one before it, without end’
and i don’t see a problem here. but if by ‘concluded’ you mean something else, i don’t understand it.
‘[…] how can addition be performed on a non-number?’ – you’re adding elements or members to a set or aggregate (jumla) i.e., the set of past infinite events. and there’s no problem with that.
regarding the third objection, most of what you said is beside the point (and contentious in any case). however numbers exist, my main claim is that Ghazali’s claim that ‘infinities don’t stand in a larger than/smaller than relation’ is false; for the real numbers are greater than the naturals, even though both are infinite.
as for this: ” Rather, that whenever you mention a natural number to some person, this person can give you a natural number greater than the one you mentioned. However, no matter the number of naturals this person lists for you, he will never be able to finish listing an infinite number of them” – it shows that you don’t understand the point at issue. i suggest you go and learn Cantor’s proof for why the real are greater than the naturals, even though both are infinite. it’s called ‘the diagonal argument’.
Wa ‘alaykum Al-Salam,
“if by ‘concluded’ you mean something else, i don’t understand it.”
Concluded as in came to an end, since the past is the series of events which leads up to and then ends with the present moment. And if the past were infinite it would never come to an end, since an infinity is endless. Thus, if the past were infinite, then an endlessness would have come to an end, and therein lies the absurdity.
“you’re adding elements or members to a set or aggregate (jumla) i.e., the set of past infinite events. and there’s no problem with that.”
The problem is that you’re treating infinity as a quantity when you claim that an infinite number of events elapsed, but then you argue that infinity is not a quantity when Al-Ghazali shows that the existence of an infinite quantity is absurd.
“However numbers exist, my main claim is that Ghazali’s claim that ‘infinities don’t stand in a larger than/smaller than relation’ is false; for the real numbers are greater than the naturals, even though both are infinite.”
Numbers don’t exist. They are abstract objects whose infinitude is a mere potentiality as has already been explained. e.g. whenever you give someone a natural, he is able to give you a greater one. However, at any given step of this counting process, the set of counted numbers will always be finite. Cantor’s arguments do not impact this fact, and there is nothing contentious about it.
I highly recommend the entirety of this presentation (although you can skip to minute 48 for a discussion on Dedekind’s and Cantor’s theories specifically), perhaps it will help make the above clearer to you:
“Concluded as in came to an end, since the past is the series of events which leads up to and then ends with the present moment.” – you use of the word ‘end’ here is tendentious, and sophistical. no series has come to an end; all that has happened is another event has occurred (which is today). if i say ‘the week ended today’, that’s just to signal an arbitrary demarcation point; it marks no real ‘end’ in the things themselves. as far as they are concerned, just another day has occurred. the relation between the infinite past and the present is like that. the infinity in question should be considered from the present back in the direction of the infinite past – for any event you choose prior to this moment, there’s another event before it.
“The problem is that you’re treating infinity as a quantity when you claim that an infinite number of events elapsed, but then you argue that infinity is not a quantity when Al-Ghazali shows that the existence of an infinite quantity is absurd.” – what i said was infinity is not any definite actual number (for any such number is always finite). but from this, it doesn’t follow that infinity is not a quantity. the denial of the specific doesn’t entail a denial of the general. (infinity is an infinite quantity, in the sense that, whatever you take from it, there will always be something left).
regarding the infinity of the past, you add to it when you consider that past as a set. so new events are added to the series of past events just in the sense that:
one event (today) occurs after another (yesterday), ad infinitum.
and then you assemble these in your mind into a set and add or subtract elements from them, depending on your purposes. there’s nothing problematic about that.
as for what comes to ‘end’, only these single events are coming to an end – in that they occur, then cease, then another comes to be, and then ceases, and so on ila ghayr nihaya. but no ‘infinity’ comes to any ‘end’.
“Numbers don’t exist. They are abstract objects […].” – just because they are abstract objects doesn’t mean they don’t exist, unless you think that anything that exists must be spatio-temporally bound or physical. but that’s just false, even on your principles.
“whose infinitude is a mere potentiality as has already been explained. e.g. whenever you give someone a natural, he is able to give you a greater one. However, at any given step of this counting process, the set of counted numbers will always be finite. Cantor’s arguments do not impact this fact, and there is nothing contentious about it.” – i don’t deny this. but you don’t seem to understand the diagonal argument, which isn’t about counting numbers and the like. it shows that it is impossible for the set of naturals to correspond to one to one to the set of reals, even though both sets have infinite members. whether or not numbers exist in your sense, this shows that, however numbers may exist, one infinite set of them is larger than another.
“I highly recommend the entirety of this presentation […].” – i listened to it. and i remain unconvinced. the author either begs the question or makes false assertions, especially his ‘divisibility’, ‘composition of units’, and ‘greater then, equal to, less than’ criteria for infinities. he basically states ‘because such operations can’t be applied to infinities in the way they are applied to finite quantities, there are no infinities’ – which is utterly question-begging.
but of course one can apply such operations, in their most basic sense, to infinities. the infinite set of naturals can be divided into the infinite sets of odds and evens for example. nothing problematic about that per se. the units of the natural can be composed or enumerated, *provided one has infinite time to do so*. nothing problematic about that. and some infinite sets are equal (e.g., the naturals and odds), while others are not (e.g., the reals and naturals), as Cantor’s argument shows. nothing problematic about that as well.
“you use of the word ‘end’ here is tendentious, and sophistical. no series has come to an end; all that has happened is another event has occurred (which is today)…”
I don’t see what’s sophistical about it. This is what we mean when we speak of the series of past events relative to any specific moment (the events leading up to and then ending with that moment, such that the last past event is the one directly preceding that specified moment). Rather, what is sophistical is your portraying of the past as a sort of potentiality when you say “the infinity in question should be considered from the present back in the direction of the infinite past”, and this is clearly not the case. Time progresses from the past leading to the present, not the other way around.
“what i said was infinity is not any definite actual number (for any such number is always finite). but from this, it doesn’t follow that infinity is not a quantity.”
But if you believe that an infinite quantity exists, then the Imam’s argument holds. Either this quantity is divisible by two, or it isn’t divisible by two. If it is, call it an “even quantity”, and if it isn’t call it an “odd quantity”… And so on and so forth.
“just because they are abstract objects doesn’t mean they don’t exist, unless you think that anything that exists must be spatio-temporally bound or physical. but that’s just false, even on your principles.”
First: if an object exists then it isn’t abstract (and just to be clear, by “abstract” I mean “I’tibari”). Abstractions can be based on some extramental existent (e.g. like many Kuliyat), but even then this abstraction would not be that existent.
Second: I know that bodies and accidents exist because I can sense them, and I know that a placeless and timeless God exists by deducing that the existence of those bodies depends on a necessary being. I do not however, have any proof for the existence of anything apart from the incorporeal God, and the corporeal entities I sense around me.
Third: when I say something like “Zayd and ‘Amr are two humans”, by “two” I’m not referring to an existent extramental entity. I’m using two as a count of the number of existents (namely Zayd and ‘Amr) to help communicate this meaning to others. But if when you make such statements you’re referring to some spaceless and timeless entity called “two”, then you will have to elaborate on what you mean, and then you will be asked to prove that such an entity exists.
“i listened to it. and i remain unconvinced. the author either begs the question or makes false assertions, especially his ‘divisibility’, ‘composition of units’, and ‘greater then, equal to, less than’ criteria for infinities. he basically states ‘because such operations can’t be applied to infinities in the way they are applied to finite quantities, there are no infinities’ – which is utterly question-begging.”
The main argument was that Cantor assumes that numbers can be arranged into infinite sets for them to be diagonalized in the first place. However, the fact that such an arrangement would entail a disparity between infinites, should have instead been proof that the arrangement itself is impossible (actually if you already agree that the infinitude of numbers is a mere potentiality, then an infinite set of numbers becomes nonsensical). And even if we accept that abstract objects like numbers can be arranged into infinite sets, that doesn’t mean that such a thing would be possible with existents in the real world. Just because one can suppose something is true, and then build theories upon those suppositions, doesn’t mean that the suppositions (or the theories built upon them) reflect reality.
salam,
“I don’t see what’s sophistical about it. This is what we mean when we speak of the series of past events relative to any specific moment (the events leading up to and then ending with that moment, such that the last past event is the one directly preceding that specified moment).”
What’s sophistical is your use of ‘end’ in the sense of ‘concluded’. Again, no series has come to an end in this sense; for there’s no series that actually exists such that you can truly say of it that it has ended. All that has happened is another event has occurred (which is today). If by the series ‘coming to an end’, you mean nothing more than what I just said – I grant it. But I don’t see anything problematic about it.
“But if you believe that an infinite quantity exists, then the Imam’s argument holds. Either this quantity is divisible by two, or it isn’t divisible by two. If it is, call it an “even quantity”, and if it isn’t call it an “odd quantity”… And so on and so forth.”
Not it doesn’t. The argument holds only if ‘infinity’ is some actual definite discrete quantity. But it isn’t that. In fact, it involves a denial of precisely that – so that our claim ‘the past is infinite’ means ‘however many (definite) past days you care to take, there are always more before that, without end’.
“First: […]. Second: […]. Third: […].”
All of this assumes the truth your materialist, nominalist ontology – which I think is false. But, because the point I make below does not ride on the truth the ontology I’m committed to, I see no need to engage the above three points you make.
“The main argument was that Cantor assumes that numbers can be arranged into infinite sets for them to be diagonalized in the first place. However, the fact that such an arrangement would entail a disparity between infinites, should have instead been proof that the arrangement itself is impossible […].”
Why? There’s nothing problematic about a disparity between infinities per se. Your claim just begs the question. Further, clearly the arrangement is not impossible in that it can be conceptualized. It may be impossible in extra-mental reality (and I may even grant it), but that’s something you have to show.
“And even if we accept that abstract objects like numbers can be arranged into infinite sets, that doesn’t mean that such a thing would be possible with existents in the real world. Just because one can suppose something is true, and then build theories upon those suppositions, doesn’t mean that the suppositions (or the theories built upon them) reflect reality.”
I conceptualize infinite sets of numbers, run the diagonal argument, and get the conclusion that the set of reals is greater than the set of naturals. Nothing problematic about this per se, And nothing about it rides on any particular ontology of mathematical objects. So it’s nonsense for you to respond to this with ‘but this doesn’t show that that’s the case in reality’ or something of that sort. Compare: if I prove to someone that ‘a2 + b2 = c2’ and they respond ‘but that’s not possible in the real world’ or ‘that doesn’t necessarily reflect reality’ or something of the like – that sort of response just betrays a failure of understanding. The diagonal argument and the Pythagorean theorem would be true if nothing but human minds existed.
There is little more that can be added to what has already been said. But it’s worth repeating that you’re treating the past as a potentiality when you say things like:
“Again, no series has come to an end in this sense; for there’s no series that actually exists such that you can truly say of it that it has ended”
Or:
“so that our claim ‘the past is infinite’ means ‘however many (definite) past days you care to take, there are always more before that, without end’.
First: […]. Second: […]. Third: […].”
And the past isn’t a potentiality. The duration that we call the past, comes to an end/concludes/finishes with the emergence of the present. And the event that directly precedes the present moment isn’t the first past event, it is in fact the last past event. This is of course, because time progress from the past leading to the present. Not the other way around. That is the key point which I feel you’re missing.
As for your claiming: “Further, clearly the arrangement is not impossible in that it can be conceptualized.”
Your ability to conceptualize a thing (“conceptualize” I assume is a translation of “Tasawur”, or understanding of what this thing means) doesn’t preclude this thing’s impossibility. For example: when the Mujasim tells me that the necessary being is actually a giant body flying around in the sky, I can conceptualize his belief (i.e. I understand what he’s telling me), but that doesn’t mean that his belief isn’t impossible.
Also a disparity between two sets of existents is indicative of finitude, and so neither set can be infinite. To use an analogy: suppose the existence of some distance Y, and suppose someone completed a journey where he traversed all of Y and then exceed it by a few kilometers (let’s call the full distance traversed on this journey X). The fact this person was able to journey beyond Y (i.e. the fact X is greater than Y), is proof that Y has a limit, such that Y ends where the difference between X and Y begins. Now suppose it was claimed that Y was actually an infinite distance. This would entail that Y be both with a limit and without one, which is a contradiction. For we have already established that Y’s being less than X entails its being limited. And an infinite distance doesn’t mean anything other than a distance without a limit. Imam Al-Ghazali’s analogy involving the orbits of celestial bodies works just as well.
Finally, you accused me of begging the question, when it is the opposite which is true. Since the existence of a set of actual infinites is precisely the matter under dispute. So you cannot assume that such a set exists, build theories upon this assumption, and then turn around and use those theories as proof for the truth of your position. That’s exactly what “begging the question” is.
salam,
“There is little more that can be added to what has already been said. But it’s worth repeating that you’re treating the past as a potentiality […].”
No, I’m not. For I deny that the (infinite) past exists at all (and so a fortiori deny that it has any sort of potential existence). Our position is one of presentism about time.
This is why you, misled by Ghazali, are attacking a straw man when you 1) assume we think the past exists actually, 2) exists as some infinite whole, and then 3) predicate ‘coming to end/terminating in the present’ of this infinite whole. Again, on our view, all ‘the past is infinite’ means is simple: for any past event you care choose there is always one before it without end.
“Your ability to conceptualize a thing (“conceptualize” I assume is a translation of “Tasawur”, or understanding of what this thing means) doesn’t preclude this thing’s impossibility. […].”
Any I never said nor implied otherwise. What I said was I can conceptualize the arrangement, even if (for the sake of argument) it’s impossible in extra-mental reality. And I only said this to show you that on your ontology of mathematical objects (i.e., where they exist only minds), the fact that I can conceptualize some mathematical object suffices to show it’s possibility – for it then exists in the mind.
“Also a disparity between two sets of existents is indicative of finitude, and so neither set can be infinite. To use an analogy: […].”
False analogy. For with regards to the real and rationals, the claim isn’t the reals have a limit beyond which the rationals go; the claim is precisely both are unlimited but that there’s more of the latter then the former – in the specific sense of an impossibility of a one to one correspondence between them.
“Since the existence of a set of actual infinites is precisely the matter under dispute. So you cannot assume that such a set exists, build theories upon this assumption, and then turn around and use those theories as proof for the truth of your position. That’s exactly what “begging the question” is.”
Now you’re equivocating between ‘real existence’ and ‘mental existence’. I, in indulging you, am claiming that the infinite sets exist only in the mind. Then, through some further operations, one of them is shown to necessarily be greater than the other, despite both being conceived as infinite. Your objection to this, as I said, is as silly as saying someone – upon being shown that ‘a2+b2=c2’ – claiming: ‘well that’s not true in reality but only in the mind’.
“No, I’m not. For I deny that the (infinite) past exists at all (and so a fortiori deny that it has any sort of potential existence).”
Yes you are. Just one paragraph later you repeated “for any past event you care choose there is always one before it without end”, and earlier you expressed this more explicitly when you wrote “the infinity in question should be considered from the present back in the direction of the infinite past”. So you think the starting point is the present, and that an infinite past merely means: for any given past event, there is one that is further away from this present moment. That’s precisely the reasoning used to justify an infinite future. The problem is of course, that the past is not a potentiality (i.e. the starting point for the set of all past events isn’t the present moment), so such reasoning won’t work.
Also: “A fortiori deny that it has any sort of potential existence” implies that if the past’s existence were a potentiality, then the past would exist. And that makes no sense.
“Our position is one of presentism about time.”
Presentism or otherwise, makes no difference. Either way, you believe that an infinite number of events entered existence. Whether or not those events currently exist is besides the point.
If someone told you “I was released after completing my seven year sentence in prison” are those words understandable under presentism? Or would you complain: “it is meaningless to claim that a duration of time was ‘completed’ if the past does not exist”?
And if those words are meaningful under presentism: was the year directly preceding this inmate’s release, his first year in prison, or his last year in prison?
“What I said was I can conceptualize the arrangement, even if (for the sake of argument) it’s impossible in extra-mental reality.”
“Now you’re equivocating between ‘real existence’ and ‘mental existence’. I, in indulging you, am claiming that the infinite sets exist only in the mind.”
When we argue that an opponent’s belief is false or impossible, we mean that this belief does not match extramental reality. I assumed this much would at least be obvious.
“False analogy. For with regards to the real and rationals, the claim isn’t the reals have a limit beyond which the rationals go; the claim is precisely both are unlimited but that there’s more of the latter then the former – in the specific sense of an impossibility of a one to one correspondence between them.”
What you described entails what was mentioned in the analogy when talking about existents. If two sets of existents are arranged so that there is a one to one correspondence between their elements, and if one set is greater than the other, then the lesser one will come to an end where the difference between the two begins. If it doesn’t come to an end, then there is a one to one correspondence between all elements, and so one set would not be greater than the other.
“Your objection to this, as I said, is as silly as saying someone – upon being shown that ‘a2+b2=c2’ – claiming: ‘well that’s not true in reality but only in the mind’.”
Don’t see why it would be silly. Just because one can invent a mental concept, doesn’t mean that this concept is actualized extramentally.
Incidentally, there are several geometric formulas that are to us only approximately true when talking about the real world. Because those formulas assume the infinite divisibility of matter, which we maintain is impossible. An example of this would be circles, and all equations associated with them. Since a perfect circle cannot exist if matter is not infinitely divisible. I suppose this would be the subject of another, much longer discussion.
salam,
“Yes you are. Just one paragraph later you repeated “for any past event you care choose there is always one before it without end”, […]. […]. Presentism or otherwise, makes no difference. Either way, you believe that an infinite number of events entered existence. Whether or not those events currently exist is besides the point.”
This, i think, fails to understand the position i’m advocating. Look, I’ll simplify it for you to the best I can.
The infinity of past events has no existence at all considered from our point of view i.e, that of the present. They are non-existents (ma ͑dumāt). Now, either you agree that we can predicate ‘came to an end’, or ‘concluded’, or ‘completed’, or some other equivalent, of such non-existents or not.
If not, you’ve no objection. But if so, then your claim i.e., that:
The whole infinite past is terminated in the present
is plainly false; for its truth requires that the infinite past be some existing (quantitative) whole. But again there was, nor is, any such whole. Rather, what existed (note the past tense) is each one of the past events, one after another. And there’s no problem with predicating ‘came to an end’, ‘terminating’, etc., of each one of these events (of the infinite past) i.e., individually.
“If someone told you “I was released after completing my seven year sentence in prison” are those words understandable under presentism? […].”
On our view, seven years is completed in the sense that each year (of those seven years) was completed, one after another. But no ‘seven year whole (quantity)’ ever existed such that one can truly say of it that it is completed.
The error in all this is the fallacy of composition.
“When we argue that an opponent’s belief is false or impossible, we mean that this belief does not match extramental reality. I assumed this much would at least be obvious.”
Which is strictly false. What we mean is that the belief doesn’t match nafs al-amr. Extra-mental reality is a sub-set of nafs al-amr.
“[…]. If two sets of existents are arranged so that there is a one to one correspondence between their elements, and if one set is greater than the other, then the lesser one will come to an end where the difference between the two begins. If it doesn’t come to an end, then there is a one to one correspondence between all elements, and so one set would not be greater than the other.”
The antecedent of your conditional argument contains a contradictory premise, namely:
The two sets (1) correspond one to one and (2) one set is greater than the other
I deny precisely that a set can satisfy both (1) and (2). So either you are saying:
1. If the two sets correspond to one to one, one of them will have an end
Or
2. If one set is greater than the other (i.e., the two sets don’t correspond one to one), one of them will have an end
Claim 1, on our view, is false. Consider: for each even number, there’s a corresponding odd number, without end. Claim 2 is precisely the point at issue. Further, I gave you an argument i.e., Cantor’s diagonal argument, to show why it’s false. You haven’t provided any argument to think otherwise.
“Don’t see why it would be silly. Just because one can invent a mental concept, doesn’t mean that this concept is actualized extramentally.”
The Pythagorean theorem isn’t “invented”; it is a truth about right triangles that is independent of how you think about them, and whether or not any is actualized extra-mentally. (Of course, in saying that, I’m speaking from the position of a realist (but a non-Platonist) view of mathematical objects. If you’re a fictionalist about them though, I don’t think I can get into a discussion about the truth between the two views at the moment. It’s a long discussion, so perhaps we can delve into it another time, leaving off this part of our exchange until that’s settled).
Wa ‘Alaykum Al-Salam,
“Which is strictly false. What we mean is that the belief doesn’t match nafs al-amr. Extra-mental reality is a sub-set of nafs al-amr.”
Missing the point. The point was: conceptualizing an idea, or maintaining some belief, does not entail the truth of said idea or belief. And in discussions with an opponent, we don’t argue about whether or not the opponent actually believes in what he claims to believe in. We discuss the beliefs themselves, since the default is assuming that the opponent is sincere.
“is plainly false; for its truth requires that the infinite past be some existing (quantitative) whole.”
No it doesn’t. There only needs to have been a last event in the duration being judged. Such that if there was a last event, then this duration is said to have come to an end.
And how strange it is of you to make such an argument, when only a few lines later you say:
“seven years is completed in the sense that each year (of those seven years) was completed”
So it is meaningful to claim that a single year was completed, but it is sophistical to claim that any other duration was completed?
“The error in all this is the fallacy of composition.”
I don’t argue: every year of the inmate’s time in prison had a beginning, therefore his entire duration in prison had a beginning.
Rather, I argue that his release from prison is proof for the finitude of the time he spent in it. For if he were required to spend an infinite years in prison, then there would be no end to his time there. Every year in prison would be followed by another year in prison, without end. But since there was an end (i.e. he was released), then this means that the time spent in prison was necessarily finite.
“I deny precisely that a set can satisfy both (1) and (2).”
Agreed.
“I gave you an argument i.e., Cantor’s diagonal argument, to show why it’s false. You haven’t provided any argument to think otherwise.”
I assumed we moved past that when you agreed that the infinitude of numbers is merely a potentiality. If this is something you still accept, then there is necessarily a one to one correspondence between all elements of any two infinite sets. In the sense that whenever someone names an element from one set, you can name them an element from the other set, without end. If this is something you don’t accept, then you will have to explain in what other sense numbers are infinite when you claim that they are.
salam,
“Missing the point. The point was: conceptualizing an idea, or maintaining some belief, does not entail the truth of said idea or belief. […].”
I never even suggested otherwise, so I’ve no clue why you’re making this point. It’s you that completely missed the point of why I corrected you on truth as correspondence: I can suppose two infinite sets, then run some further (mental) operations on them, and infer the conclusion that one set is necessarily larger than another (i.e., Cantor’s argument). This would hold true even your physical world of atoms and accidents never existed. In other words, corresponds to nafs al-amr but not to anything (spatio-temporally) external.
“No it doesn’t. There only needs to have been a last event in the duration being judged. Such that if there was a last event, then this duration is said to have come to an end.”
What’s this “duration” you speak of? The infinite past? If so, you erroneously treat it as some existing quantitative whole and so commit the fallacy I mentioned. Or do you mean the last past event i.e., yesterday? If so, to say the “duration came to end” is then just to say “yesterday came to end” i.e., yesterday’s duration came to end. But nothing problematic about that.
“So it is meaningful to claim that a single year was completed, but it is sophistical to claim that any other duration was completed?”
What’s sophistical is treating the infinite past as some wholly existing duration and then claiming that it is completed as some infinite whole on the basis of each one of its successive members being completed.
“I don’t argue: every year of the inmate’s time in prison had a beginning, therefore his entire duration in prison had a beginning.”
No, you argue: every past event of the infinite past was completed, therefore the whole of the infinite past was completed.
“Rather, I argue that his release from prison is proof for the finitude of the time he spent in it. […].”
Sure, but nothing here to help your case. Every day I claim is followed by another day, without end. And every past day you like was preceded by another day, also without end. There’s no equivalent in the infinite of past events case to your ‘released from prison’ in the analogy you have given.
“I assumed we moved past that when you agreed that the infinitude of numbers is merely a potentiality. […].”
You just don’t seem to get the argument. Whether potentially or actually, the diagonal argument goes through. Look, let me simplify: take the real number 1.23462456 (I just randomly generated that). Give me the natural number that you think corresponds to it.
Wa ‘Alaykum Al-Salam,
“I never even suggested otherwise, so I’ve no clue why you’re making this point”
This all started as a response to: “the arrangement is not impossible in that it can be conceptualized”.
You made a similar statement in this most recent post when you argued: “I can suppose two infinite sets, then run some further (mental) operations on them, and infer the conclusion that one set is necessarily larger than another (i.e., Cantor’s argument). This would hold true even your physical world of atoms and accidents never existed.”
But conceptualizing an impossibility is not impossible, and supposing an impossibility is not impossible either. So claiming that you can conceptualize your beliefs, or suppose their truth, is not enough to prove that they aren’t impossible.
“What’s this “duration” you speak of?”
The sequence of events being judged.
“The infinite past? If so, you erroneously treat it as some existing quantitative whole and so commit the fallacy I mentioned.”
How? I did not assume that all past events concurrently exist, nor does what I argue for entail this (at least you never show that to be the case, you just baselessly assert it). All I’m saying is that each of those events entered existence, and that there existed a last event for this sequence. This is enough to prove that the sequence necessarily had a beginning.
“What’s sophistical is treating the infinite past as some wholly existing duration and then claiming that it is completed as some infinite whole on the basis of each one of its successive members being completed.”
You completely avoided dealing with the objection I raised. Let me spell it out for you.
You earlier say: “each year (of those seven years) was completed”, even though the single year is a duration composite of many events that didn’t concurrently exist (successive events that make up months, weeks, days…etc.).
At the same time, you argue that it is meaningless to claim a duration was completed, because the events that make up this duration don’t concurrently exist (and for some reason, you think the statement “this duration came to an end” entails “the events that make up this duration all existed simultaneously”).
Given the above, either your statement: “each year was completed” is meaningless. Or, when you state that you don’t understand what it means for a duration to be completed, you’re claiming to not understand something which you actually do understand. And if the latter were not pure sophistry, I don’t know what else would be.
“No, you argue: every past event of the infinite past was completed, therefore the whole of the infinite past was completed.”
The past was completed by definition of being the past. After all, what was not completed is not in the past.
Also: didn’t you yourself commit this fallacy when you said: “seven years is completed in the sense that each year (of those seven years) was completed”?
“There’s no equivalent in the infinite of past events case to your ‘released from prison’ in the analogy you have given.”
The analogy was provided to help you realize that if a duration came to an end, then this duration necessarily must have had a beginning. And the sequence of events that makes up the past has an end, therefore this sequence must have had a beginning.
“Whether potentially or actually, the diagonal argument goes through.”
No, it makes a huge difference.
If you believed that numbers don’t exist in of themselves and that their infinitude was merely a potentiality, when you say “1.23462456” you would be asked to explain in more detail what you mean. 1.23462456 what? 1.23462456 meters? 1.23462456 cakes? 1.23462456 hours? 1.23462456 as a term in a counted list of reals?
If you mean 1.23462456 as a term in a counted list of reals, then when you ask “give me the natural number that you think corresponds to it” this is taken to mean: if I start listing real numbers, and I include 1.23462456 as a term in my list, what is the position of this term in that list? And there will always be a valid answer to that question (since you already agreed that at any given step of the counting process, the set of counted numbers will always be finite).
But of course, that’s not what you mean when you ask: “give me the natural number that you think corresponds to it”. Rather, your question assumes the existence of a world of mujaradat, wherein an infinite number of numbers actually exist. And you believe those mujardat can be signified with expressions, that when arranged into a table, can be used to prove that one infinite collection *of those mujardat* is greater than the other. And so you believe that there actually are an infinite number of reals, and that one of those actually existent reals is signified by the expression “1.23462456”, and that this real cannot correspond to any member of the actually existent collection of naturals.