Well, while that is the general nature of premises in an argument, it’s also the case that if one didn’t feel any reason to accept Aquinas’s premises, they would not be compelled by his argument to feel any reason to accept his conclusion. I think that’s mainly all people are pointing out in complaining about the premises here; that Aquinas’s argument is, for them and/or others in general, unproductive, if nothing else.
Sez you. Suppose I’m already of the mind that the universe could “progress through” an infinite number of events in a causal chain. What have you offered to dissuade me from this?
Incidentally, we could easily have a set-up where there is a chain which is infinite in total extent, but where any two links in that chain are only finitely separated. [E.g., consider the natural numbers or integers; infinite in extent, yet there are only finitely many between any two]. But I’m a smarter man than to waste too much time drowning myself in that morass again…
That is all well and good, but without any logical reason to deny his premises than I would suggest that they are simply not engaging with argument. And if the argument is unproductive, is that the fault of the argument if you refuse to engage with it? How would you suggest someone responds to a criticism of “I don’t feel like accepting your premises”? It is not a position based on logic, but really on emotion.
Calculon.
Do you disagree with the premise that:
-infinty + n = -infinty
Can you give me any generally accepted case in nature where an infinite set has been progressed through? Not just in maths thought experiments but in real everyday experience?
Calculon.
“Number” is an overloaded term. There are axiomatic definitions for the different classes of number, which a quick Google search can show you.
The slope of a vertical line is undefined, not infinity. A slope is the rise over the run, the run in this case is 0, and so it can be found only by dividing by 0, which is of course undefined.
It needn’t be anything other than a position based on the fact that the position-holder doesn’t feel any reason to accept the premises, nor have they been given any. If you want logic to push people towards accepting the premises where they did not already see cause to, you’ll need to use logic to reduce to these unconvincing premises to other, more manifest premises. If at some point you find you can no longer do so, you’ll just have to accept that perhaps some won’t feel any reason to grant the premises you’re stuck with.
I can prove anything I like, given sufficient premises. If you don’t already have some reason to suspect those premises of being true, why should you care overmuch what I am able to establish as conclusions from them? Sure, the abstract entailment structure of various propositions may be of some interest from a logician’s perspective, but if all you are concerned with is whether the end-conclusion is compelling or not in itself, then I doubt you will be terribly impressed by arguments from arbitrary premises.
To put it in concrete example, consider the argument
Premise 1: There is life on other planets
Premise 2: Carbon-based life only exists on the Earth
Conclusion: There exist forms of life which aren’t carbon-based
Supposing one were to say to this “Meh. I’m not convinced. I still think life has to be carbon-based, as I don’t see any reason to accept those premises”. Would that be an unreasonable response to make?
Yes, I agree. That is my point. Overloaded and continually overloadable, with no particular formalization having distinguished canonical status.
Also true. And that same quick Google search will show that many of those classes include numbers construed as “infinity” or “infinite”. Not the integers or the reals, of course, but other ones just as worthy of discussion.
It’s undefined in a system where division by 0 is undefined… in other systems, it’s not. I chose that example because it is the precise motivation for the projectively extended real numbers.
As I showed a while back, addition is not defined on infinity, it not being a real number, so the equation is meaningless. I don’t care about the infinite regress argument, but your math must make sense.
This all depends on the purpose of his argument. If he is making a purely philosophical one, we can examine if the conclusions stem from the premises, without concern for the accuracy of the premises, as is often done in math. If however the conclusions are supposed to say something about reality (god exists) then the premises must be tested against evidence. In this case they fail badly.
It depends. What sort of thing is n? (For example, is it required to be finite, and, if so, why?). What particular mathematical system of arithmetic are we working in and how does it apply to the issue under consideration?
You speak as though you have the odd idea that addition is only defined on real numbers. 
A better response may be the argument to show that premise 2 is false, namely:
Premise 1: The laws of nature are the same everywhere
Premise 2: There are many earth-like planets in the universe, and as such the earth is not unique or special
Conclusion: Any life that exists on earth, including carbon based life, can exist elsewhere.
That conclusion of that argument invalidates premise 2 of yours, and so the argument is mis-formed at best.
Calculon.
That may be a better response. But maybe the responder isn’t smart enough to come up with that instead. Or maybe, for some reason, it doesn’t apply (perhaps God visited the responder and told him Earth actually is special, though God declined to name the details). Or maybe I’ll change the example slightly…
But why bother? At any rate, I’m not concerned with the question as to whether there are better responses. I’m just concerned with the response I actually asked about. In itself, would it be unreasonable or not?
[Fine, let’s toss out a new example.
Premise 1: Indistinguishable has an older brother
Premise 2: Indistinguishable has a younger sister
Premise 3: Indistinguishable has two siblings
Conclusion: Indistinguishable is a middle child.
Are you convinced that I’m a middle child now?]
n is a rea lnumber. The ultimate question being can you bridge an infinite gap with finite steps? If the answer is no, then the universe could not progress through finite causes through an infinite number of causes to now.
Calculon.
(More to the point, if you aren’t, is that evidence of obstinate refusal on your part to properly engage with my argument?)
How does “Can the equation -infinity + n = -infinity fail for n a real number?” apply to the question as to whether the universe can “progress through an infinite number of causes”? For example, what in the latter demands the restriction to real numbers in the former?
If you can’t show that one of the premises of an argument is false, then I think it unreasonable to say that the conclusion is false. I also think it unreasonable to say that someone has not formed a good argument because their premises are unproven, but not necessarily false. I guess one could remain agnostic, in that you neither believe or disbelieve the argument. However if you have no reason for saying that the premises are false then it seems a position based on emotion and what you wish to be true, rather than on logic.
Calculon.
Well, if you accept the validity of the entailment from the premises to the conclusion, yet have reason to say that the conclusion is false, that in itself will be a way of showing that the premises of the argument are false. (“One man’s modus ponens is another’s modus tollens”…)
E.g.,
Premise 1: Indistinguishable has an older brother
Premise 2: Indistinguishable has a younger sister
Premise 3: Indistinguishable has (precisely) two siblings
Conclusion: Indistinguishable is a middle child.
Suppose you have some separate reason to suspect that I am actually not a middle child (you can concoct a story for yourself; the details don’t matter). You are now in a position to say that the conclusion of the above argument is false, even though there aren’t particular premises which you are in a position to show false, and, indeed, your only reason for believing “At least one of those three premises is false” is the very fact that you believe the conclusion to be false. And yet, is this not an entirely reasonable stance for you to take? Why should the above argument have caused you to think I was a middle child, if you had no reason to accept its premises in the first place? You could hardly be blamed for not giving it much stock.
Not entirely. I posit another argument:
Premise 1: You are trying to make me look foolish (no need, I do a pretty good job myself 
)
Premise 2: Having me agree that one your premises is true when it is not would make me look foolish
Conclusion: One or more of your premises are not true
To be certain I would first have to trust that your account of your family is accurate. I hope it is, and that my argument is wrong. But still I can posit an alternative arguement without having to resort to just saying “I don’t believe”
Calculon.
This is an important question which I think needs more emphasis. Mathematical structures do not alter or create reality: if you use mathematics to describe natural phenomena, you also have the burden of demonstrating that the universe behaves in a way that is reasonably well described by your models.
So, Calculon, “Infinity + 1 = Infinity” may hold in some mathematical framework that you choose to employ, but saying “Infinity + 1 = Infinity, therefore an infinite chain of causality is impossible” is missing a whole slew of steps in which you must establish that the universe is actually bound by some phenomenon that agrees with your math, in addition to showing how your notion of infinity as a quantity implies the restriction that you claim.