For mathematics in general, he began with five axioms, fundamental truths that could not otherwise be proven via mathematical argument. When he got to geometry, he added five additional postulates--which are also axioms for Euclidean geometry--as the basis of his system. To whit, he postulated the following must be possible:
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any center and distance.
4. That all right angles are equal to one another.
5. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.Note how the first four are short and precise, unlike the fifth postulate (also known as the parallel postulate). Various mathematicians since Euclid have attempted to "clean up" the parallel postulate, with limited success. The problem is that it is simply not as intuitive as the others. Euclid knew this; the first twenty-eight theorems he derived in his geometry were based wholly on the first four postulates. Indeed, there is a geometric system--called absolute geometry--that is limited to only the first four postulates and the theorems that can be deduced from them.
And there are also other systems--called hyperbolic geometry and elliptic geometry--based on a different fifth postulate. One of the ways of rewriting the fifth is the following (the Playfair postulate):
Through any given point can be drawn exactly one straight line parallel to a given line.If we allow that, instead of one parallel line, more than one parallel line could be drawn through the given point, we end up with hyperbolic geometry. If we allow that no parallel lines could be drawn through that point, we end up with elliptic geometry. Thus, these two geometries have come to be known as non-Euclidean geometries, since they both violate one of Euclid's postulates (absolute geometry is not non-Euclidean, since it doesn't actually violate the parallel postulate).
What is significant about the existence of these geometries is that they are fully functional; the same theorems that are true for Euclidean geometry that don't require the fifth postulate remain true in non-Euclidean geometry. And drawings can be made, shapes constructed, and problems--including real-world ones--solved with these geometries.
There's a lesson here, as well. For centuries upon centuries, Euclid's postulates went largely unchallenged. There was no geometry but his; to suggest otherwise was near-blasphemous, a strange thing in the world of science. Thus, Euclid's assumptions came to be taken as wholly foundational: the five postulates for his geometry were accepted as absolute truths; there were not merely theoretical truths, they were truths of nature, of the physical world as well. They were--to return to the Greek world and Aristotle--a priori truths, requiring no empirical evidence for verification. And all of the theorems subsequently derived from them were the same. In total, they were taken to be the only rules that actually could and did describe reality.
We now know this is false. Reality is not dependent on such rules in the least. Some two thousand three hundred years after Euclid, Kurt Gödel would demonstrate that our traditional number theory was also not so absolute--as mathematicians and scientists had come to believe it was--via his incompleteness theorems. Simply put, Gödel showed that any sufficiently large--as to be useful--axiomatic system contains true statements that are not provable by the rules of the system. To put this another way, Gödel proved that our system of mathematics was incomplete, and necessarily so. It cannot provide all the answers, no system of mathematics can, and this must always be the case.
With that in mind, I'd like to talk about the basis of our common morality, with respect to some ideas we--as humans--tend to assume as foundational truths. Philosophically, what we're talking about here is the field of ethics. And in that regard, there are all kinds of ethical systems to look at, developed by various thinkers in various cultures throughout human history. I have no intentions of wading through this plethora of systems, but I would not that what differentiates one from another is usually organization and consistency: the most notable ones are those that are well-organized, that endeavor to remain consistent.
And that's just common sense, I think. Such things do not validate the conclusions of a given system, but they do lend a formal--even a scientific--air to the system. Still, all of the systems--save a small handful--do share something: first principles, of one form or another. But what justifies such principles, themselves?
Immanuel Kant--for instance--formulated his famous Categorical Imperative on the basis of assumptions about what it meant to be human, what humanity entailed. Suppose one or more of those assumptions is not necessarily true? What does it do to the Categorical Imperative and the resulting axiomatic system that can be theoretically derived from it?
Utilitarianism is based on a simple rule (the principle of utility): the correct action or choice is the ones that leads to the most happiness for the greatest number of people. But implicit in that rule are assumptions about happiness and how people perceive it: it is assumed that different individuals will understand identical consequences in the same way. But is this true? Is it necessarily true? If not, what are the consequences for a utilitarian system?
The point is, a given ethical system, no matter how elaborate and well-conceived, proceeds from assumptions about man, about good and bad, about justice and injustice, that may not be the only possible assumptions. Moreover, its systematic nature is, itself, problematic in this regard, as the more elaborate it is, the more likely there will be truths (with respect to first principles) that cannot be ascertained.
The counter to systems of ethics is most often religion: no need to establish a means of determining ethical behavior when there are religious tenets explicitly defining good and bad, right and wrong. The criticism of such tenets is simple: who say so? If it be a deity, what if that deity doesn't actually exist? And such criticism has a fair point. Ethical systems based on religious tenets are based on an overarching assumption: that the tenets are true, as a matter of course.
And here's a thought: wouldn't such an approach--simply telling people what to do--likely serve the principle of utility more effectively? Greatest good for the greatest number, after all...
Still, there is liberty and freedom of choice to consider. These can be taken as first principles, as well. But the basis of the latter is surely unassailable: we can always choose. Always. If not, there would be no need for a system of ethics, at all.