I'm highly skeptical of this modern method we have of citing science to dismiss ideas, without actually having science behind our citations. It's great to have the
idea of cells, chemical reactions, and the modern outlook that math must explain everything.
But I know from my own studies that there really isn't a mathematical equation for this sort of argument -- only a mathematical viewpoint, and that is an entirely different animal.
Show me the equation that has chemicals come in with the reactants and life come out with the results. If, as you both suggest, life is
common, how come we can't go study the first formation of life? Not from pre-existing systems, with eggs and sex cells, but where no life previously crawled.
I
know single celled life
can form. My arguments (on both sides) must rely on a few basic assumptions, or I won't be able to make them at all. Since this is for a project of mine, I'd like that we all regard them before making a statement--unless, of course, they have something provable that undermines one of the assumptions.
(1 - That randomly formed life is so rare so to be mathematically insignificant.
(2 - That infinity as a concept is different from the reality of the universe.
(2a - because otherwise, wouldn't infinite space and infinite time also allow for infinite matter? And since Infinity = Infinity, infinite matter would fill infinite space, which must mean that infinity as a concept and the reality of the universe are different things.
(3 - That God (or god, or G-d

or a designer) is a possibility, not a positive statement that must be proven true or else be assumed false.
(3a - The entity or phenomena referenced in point 3 is not necessarily God. Nor Allah, nor a Mystic Law, nor the main supernatural power of any specific doctrine.
I have an idea, but it doesn't have it's own word, yet. I'm also afraid that my idea might be as offensive to Atheists as it would be to Christians, so I'd rather not say it at all.
Good post. You say you're not okay with using science and math to dismiss ideas, and then your point #1 is using (weak) math to dismiss an idea. And in a very judgemental way: you decide the event is mathematically insignificant, then use that to belittle the idea. Which is okay, since your "randomly formed life" is a weak concept anyway. Chemistry is not random. As I said, you can make complex chemistry seem nearly impossible by trumping up the odds, but if you're familiar with chemistry the vast majority of "random" possibilities don't happen, or they're short-lived enough to not be a real factor in a long-term outcome. Sure, which ions bump into each other at any given instant are unpredictable, but the outcomes of each reaction are very limited.
In any case, it's not honest to dismiss the known natural processes as being too unlikely for your mind to handle, and then substituting a designing intelligence as being more likely. If you use probabilities on one side, you need to apply probabilities to the other, and let the more likely one prevail. Although sometimes the less likely answer turns out to be right. Probabilities alone do not lead to answers.
If you think chemistry is hard to pin down, try spelling out what properties this nebulous "designer" must have. My problem here is that the "designer" people can't even say what it is they're looking for. Some kind of lifeform that lived before/beyond the universe, has a lifespan of billions of years, is somehow responsible for setting up the laws of nature, if not molding the earth personally and wishing to nurture us ... you really believe such a thing is out there? I see these probabilities: a creature living beyond/before the universe = 0, a creature living for billions of years = 0, personally molding a planet = 0, hearing the prayers of billions of life forms = 0. Feel free to counter these estimates with better data. I'll stick with known chemical processes any day.
Yes there's a critical point where chemistry became life. We haven't found it or reproduced it yet. The designer folks put that in their column automatically, but that's not valid. Their theory must be that some divine force put the spark there -- in what, the first cell? the first billion cells? And what was this spark? DNA? A soul? Do cells even have souls? They can't define it any better than science can. They're just counting on invisible things. Be honest, the lack of an answer supports neither case. Don't forget that there are also things like viruses that we're not sure whether they're alive or not, and only a tiny fraction of early life forms were preserved in any way. It's not as simple as it sounds. That we've learned so much is admirable. Criticizing the whole process for not having found every answer by the year 2009 is invalid.
To say that math doesn't explain anything is also shaky. Mathematical models are part of each theory, and when they work, they persist. You say, "there really isn't a mathematical equation for this sort of argument" -- but this is vague. Theories are not just mathematical equations, and if a theory uses calculus to predict the motions of real objects, it doesn't mean that we need to start looking for integrals floating around in the ether. Some mathematical operators are abstract tools that act on real quantities. Taking the math too literally is invalid. And what is "this sort of argument"? In any case, don't forget the alternative: there's no equation for the supernatural either. If you somehow undermine all of science (which you won't) it doesn't support any particular alternative. I'd say this is the biggest flaw in the religious quest: the infuriating notion that poking holes in someone else's work supports your own work.
Your #2 point, quibbling over infinity is pointless (in this context). Though it's okay to point out that when people casually say space is infinite, they're talking about a layman's infinity (simply beyond reckoning) as opposed to a strict mathematical infinity. It doesn't change the odds. Anyway, your point 2a is flat-out wrong. Infinite space would "allow infinite matter", but that doesn't mean inifinite matter MUST result. Even if it did, there are different sizes of infinity, and one infinite thing could contain another. Your "Infinity = Infinity" shows you don't know much about the math. All that is missing the point anyway. Even if you were right about your idea of infinity, it has nothing to do with the present argument. And, the existence of that layman's infinity, various mathematical infinities, and phrases like "effectively infinite", if people don't say which infinity they're referring to, the result is meaningless.
Anyway, I can't really credit any of your talk about math and experience, since you misuse math and say nothing about that experience.
Point #3. Things can get touchy here, but we've been pleasant about it. Sure, God is a possibility, except that it's a slippery invisible one that, every time you look someplace, gets relocated somewhere else. And that really doesn't make a good case for itself. People used to say storms were caused by gods -- is that still the case? Or is it just that anything we don't personally understand gives us that vague uneasy feeling which we need a name for? The modern trend in this search is that anything not explained by science must be evidence of higher powers ... but, no, it doesn't have to be any such thing. Every time this comes up, it's invalid. This notion of "you're wrong so we must be right" is childish, and just won't fly. Folks on the supernatural side just keep saying things that are baseless, meaningless, or off topic, and are unaware that the arguments have no weight with rational minds. It's frustrating all around.
The reason science rejects these notions is that they're not scientific (not logical, not rational, choose your term), they're just feelings and hopes. It actually gets annoying after a while. It's like mopping a floor and having someone stand behind you saying "You missed a spot, you missed a spot." One side is doing real work, and the other side is complaining that it's not coming out the way they want it to. The floor is essentially clean. The few missed spots don't automatically support supernatural forces.
(3a) I agree that the "designer" is no longer one of those conventional deities from ancient time. Yet it's an underground version of the same thing -- the meaning of the universe, the supernatural, the answer to so many prayers. It can't help retaining that emotionally charged baggage. Its seekers just keep renaming it, trying to fool someone. Sorry, but to the rational crowd it's the same old thing.
To say you have a word for it that nobody else will tolerate is a lame dodge. Either say what's on your mind or don't bother. The odds of it being original in the annals of humanity are slim. In any case, it would be nice if the designer crowd could decide what it is they're looking for, instead of lumping together gaps and invisible stuff and thinking they've figured something out. I keep encouraging them to come up with a more solid theory, but there's never anything solid to work with. They just keep throwing bits and pieces in a folder and hoping a tree will grow. Until then, I will continue to reject smoke and mirrors and intangibles, and ask for better arguments from your side. If you're suggesting that I should be okay with a jumble of shifting nonsense that can't be seen, sorry.
Now I admit that there's a vague "feeling" in the world, that all things are connected somehow. And that it would be comforting if there was a purpose to it all. Still, neither one is a valid data point, and they feel to me like projections of ourselves onto the world, not part of the world itself. In any case, unless we can prove those impressions come from outside of us, they're inadmissible. Another interesting field of discussion.
Thanks for listening. I hope this gives you some writing ideas that may help with your project. My main aim is to show you what those theories are up against and how little water they hold.