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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.
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?
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 28

[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.

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.