What follows is a dilettante's guide to the history of Enlightenment, stopping off here and there to visit some Famous Names or Results in the development of philosophy. Modern philosophy in the West went from a state of near-complete religious domination (where nearly the only philosophers acknowledged to exist at all from the times of the Greeks were religious ones such as St. Anselm or St. Thomas Aquinas (and look what their philosophical musing got for them - sainthood!). The Church more or less endorsed much of the philosophical musings of the Greeks because they were free from the taint of religion and ``safe'', and provided the illusion of answers to many non-God questions.
We will skip at this point the various details of the work of (in no particular order) Francis Bacon (arguably the father of Natural Philosophy and all of modern science), of Kepler and Tycho Brahe, of Hobbes and of Galileo - all contemporaries of (on the early side - they were all a bit older but overlapped) Descartes. However much I enjoy telling this particular story in physics class (because they are all relevant primarily to the development of natural philosophy as I discuss later) they, and Copernicus, and the countless alchemists, and Columbus, aren't immediately relevant to my thread of examining axiomatic, deductive reasoning and the real world.
We'll skip ahead to the philosopher who might well be called the father of rationalism (so much so that it is properly referred to as Cartesian rationalism), a monster mathematician as well as a philosopher, to one of the first men to attempt to apply the methods of pure reason to the Really Big Questions such as the SUW.
I refer, of course, to Renee' Descartes.
Descartes' personal life is of little interest to us here. He was French. We was born in 1596 and died in 1650 (when a man named Isaac Newton was seven-plus years old). His family was well off (the sine qua non of early philosophers was independent means or at least sufficient wealth from a real job that it provided them with liesure time for philosophizing). He was educated by the Jesuits.
Somewhere in there he learned that he was a genius with an aptitude for mathematics and focussed his energies on mastering mathematics and philosophy. He wrote a Theory of Everything, but (possibly out of fear of the Church, who had just whomped Galileo upside the head with the threat of a stake piled with firewood for daring to suggest that anyone with eyes and a telescope could see that the Earth was Not In The Middle) he abandoned it unfinished. He then went through with writing and publishing several works and appendices on both mathematics and philosophy, which are generally unexciting except for three things - he invented coordinates systems for the first time providing a quantitative frame for geometry; upon this rock he invented analytic geometry, a prerequisite for the eventual development of physics and the calculus by Newton, Leibnitz and others, and he wrote cogito, ergo sum (I think, therefore I am) as the basis of his attempt to prove using pure reason (of which he was inordinately fond, possibly because he was so damned good at it) that everything is the way that it is because it must be, because there is some rational basis for it all.
He didn't just want to understand reality or prove the God exists using something like the Ontological argument13.2, he wanted to derive it.
To help us forgive Descartes, note that he worked in a world where the Church still had a habit of really hurting those that opposed it (at least in their perception), and where the population of the world was small enough that we tended to have only one or two geniuses born into the appropriate social class at any given time. Doubtless there were other polymaths alive out there (other than the not-so-mathematical folks on the short list above), but they probably were tending goats on some hillside in Spain or Italy wondering why goat hair made interesting patterns when it was being rained on. Also, scientific communications were slow and what now might take days to report to the world then took at least weeks and sometimes (where the New World was concerned) years.
To give true and proper credit to Descartes, in the process of reinventing geometry for the first time since its original invention, he fell in love with the very concept of axiomatic reasoning, and his work necessarily preceded the serious reanalysis of the axioms of Euclid and the discovery that they were flawed in many ways13.3. To him axioms were still, to a great extent, self-evident truth, but he nevertheless knew full well that he had to work from axioms of one sort or another.
So Descartes set out to determine what he ``knew'' with absolute certainity, so that he could use this as the axiom set from which to proceed in his reasoning process. The axioms he sought and adopted were very much of the ``self-evident truth'' variety, because he wanted his conclusions to be as well-founded logically as those of Euclid - he wanted a veritable geometry of Being, including a theorem of Deity (as he was a profoundly religious man13.4).
So he proceeded to figure out what he could doubt. If something was dubious or doubtful, it couldn't be an axiom, right? When he was done, what was left would become the axioms of his rational system.
Descartes rapidly realized that when you got right down to it, there was damn-all that one couldn't doubt when one tried hard enough. Can I doubt that the sun will rise tomorrow? Certainly! Maybe it will, and maybe it won't, we can know nothing of what hasn't happened yet. Can I doubt that I ate lamb for dinner last night? To be sure, everyone knows that human memory is fallible. Perhaps I ingested large quantities of hallucinogenic drugs last night and just fantasized that I ate lamb for dinner. Perhaps an Evil Fiend hypnotized me into believing that I ate a dinner of lamb. Perhaps I was rendered unconscious in my sleep for exactly one day, and I actually ate lamb two nights ago but haven't figured it out yet. Perhaps what I ate was mutton, dressed up like lamb.
And so it goes. Can I doubt what I am seeing? Surely. Again, hallucinogens, particularly vivid dreams. Look at Keanu Reeves in The Matrix, moseying along thinking in perfectly good faith that he was a computer programmer living and working in a clean, urban world, only to wake up and discover that he was only a power unit in the Matrix (and later discovering that the world in which the Matrix resided might well itself be in a higher order Matrix, starting off a potentially infinite chain of layered meta-realities that would gladden Hofstadter's heart). That which I see can be imagined, that which I hear can be projected, that which I feel or touch can be simulated. The external reality that I believe these sensory impressions correspond to could be real, or they could all be some sort of metaphorical Matrix. They could even be both!
The past is thus uniformly dubitable - perhaps I was created moments ago with memories of the past intact. Could I tell? No - the memories I have don't even always correspond to other people's memories of the same events13.5. The future is dubitable as I haven't even the continuity of memory to help me with that which has not yet happened. Most of the present is dubitable, because I know it only via my senses and they are not trustworthy.
The one thing that I know, that I cannot doubt, is that I exist. I cannot doubt my own existence unless I exist. To think, to doubt, even to dream is to exist. So it is with me. So it is with you, as you think over these words. Cogito, ergo Sum13.6. And so it was, I imagine, with Descartes.
Descartes now had his one irrefutable truth, his Axiom, and proceeded to try to derive Everything from it. For a good mathematician, he botched the job horribly.
His reasoning went as follows: I exist. My existence must have a cause. My existence cannot be its own cause because it just isn't up to snuff - my awareness seems to wax and wane with sleeping and waking, I have been unconscious altogether, my existence seems altogether ephemeral and insubstantial when viewed as a self-sufficient cause, so I must have a cause ``greater'' than myself13.7, since I'm so obviously flawed and imperfect and not-so-great.
Hmm, cannot be anything in the world I see (historical evidence to the contrary, sorry Mom and Dad... you're just dubitable) as this just puts us onto one of the pathways to the Pit of Existential Despair as even if I use ``the laws of physics'' as a Prime Cause they just aren't great enough. Indeed, none of this is any better than I am at causing things without itself needing a cause, assuming that it is there at all. Even if my cause really IS something my mother and my father did one night long ago, they needed a cause, and that cause needed a cause, each cause greater than the one before. The whole world would need a cause. The whole Universe would need a cause. Physics would need a cause. Must be something greater than me and the whole existing Universe and Physics besides (if either one exists, of course, which I find dubitable)!
I know, let's call this Greatest Cause, this prime cause, a Guardian13.8, oops, no, I meant God. As the Greatest Cause it clearly must be perfect and good and all-powerful and all-knowing and all-self-referential and everything like that because (shades of ontology) if it wasn't I could imagine a greater cause that was more perfect on to the limit of infinitely perfect (Descartes with his number line almost certainly had a nice appreciation of infinity). Surely this perfect prime cause, this God wouldn't cause me to exist (the one thing I cannot doubt) but fake me out with a non-existent Matrix because that would be a lie! So the world I see is real, it exists too! Hooray, I don't have to be a solipsist! My memories must be real13.9! I've just proved the logical necessity of God and All Observed Creation from just one itty bitty Axiom (in the self-evident truth sense), the indubitability of my own existence!
Ain't I cool...
Not. Descartes made mistakes by the fistful. His biggest was in not listing all of his axioms and the fact that axioms are by definition (as assertions) dubitable. Also, cogito, ergo sum is not an axiom - it is something different, something that the world had only very, very begun to realize was the basis of a new kind of conditional knowledge, one that actually worked. It is an empirical observation. More on this later.
Descartes' entire discussion is thus a subtle form of the fairy argument - by the time you start arguing over whether or not you require a greater cause than yourself or not you've already lost the battle for pure reason no matter what you decide. Let's help him out and write down (and doubt) a bunch of his unwritten axioms, starting with the show stopper:
We now have axioms for something called ``causality'' in physics that works the way causality should, intuitively, work. There is no greater than or less than inherent in it - all interactions between all particles in the Universe are equal in greatness as regards quality of ``cause'' (the interaction) and differ only instantaneously and dynamically with regard to quantity of cause (the strength of the resulting force). Physicists don't look down on Gravity, for example, just because it is all wimpy and everything compared to the Strong Nuclear force.
The argument is at best a conditional argument, then, which should come as no surprise at this point as all arguments are basically conditional upon the premises and axioms and reasoning method used. It is further flawed by its use of self-referential constructs - all questions about God that one attempts to answer via any sort of ontological condition are intrinsically self-referential as they require comparisons of a class that is, by definition, incomparable. Descartes most basic argument is that because he remembers (with his possibly false memory) that things within nature appear to have causes, that causality itself (as expressed in the laws of nature) must have a cause. This is applying set-theoretic rules not to objects within the sets but to the sets themselves as objects in the theory, looking for (quite literally) the set of all causes that are not in the set of things that are caused.
We have seen in fair detail that this makes the conclusion to this sort of reasoning null even aside from the conditional nature of his premises - it isn't just possibly correct or possibly incorrect, it is the class of reasoning that can lead to undefined or null conclusions - wrong at the level of a Gödellian knot.
It is Bullshit.
Note well that bullshit or not, Western Civilization is built on top of science which is built on top of Cartesian Rationalism. There is much of Descartes' arguments that one might want to adopt as part of your set of prime axioms, as long as in fairness you recognize that you can't prove them and that his argument in terms of God as a Prime Cause is interpretable at best as poetry or statement of faith, not as logic. It is perfectly reasonable to believe in God. It will never be rational.