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As a brief introduction to mathematics, I have here a short video about Pythagoras.  Special thanks to S. Peter Davis for making it.


More about the Pythagoreans 

 From Aristotle,

But the Pythagoreans have said in the same way that there are two principles, but added this much, which is peculiar to them, that they thought that finitude and infinity were not attributes of certain other things, e.g. of fire or earth or anything else of this kind, but that infinity itself and unity itself were the substance of things of which they are predicated.  This is why number was the substance of all things.  Aristotle, Metaphysics, 987a


The important part is finitude and infinity (i.e. numbers) are not attributes (or characteristics) of certain other things.  For example, "Threeness" (a universal), is not a characteristic of "those three balls," (a particular), but rather it is the other way around;  "threeness" is the primary substance which is predicated by instances of three ("those three balls").   They believed universals are real and not their particular instances of it.


Regarding Geometry


                                        1 is the point,

                                        2 is the line,

                                        3 is the plane,

                                        4 is the solid.


 These numbers add up to ten.  (i.e. 1 + 2 + 3 + 4 = 10)  As such, they called 10 the Decad or the perfect number as it contains all the elements of number.  Our base ten number system is founded on these ancient beliefs.




Democritus of Abdera




Over two thousand years ago in Greece, there was a really brilliant philosopher and mathematician named Democritus of Abdera who lived about 460 BCE.  He was said to have been incredibly talented in geometry and we have enough second hand accounts that indicate he made advancements in number theory,  mapping, tangences and irrationals.  Most of what he wrote did not survive the middle ages, but below are three fragments of his that did survive.





 If a cut were made through a cone parallel to its base, how should we conceive of the two opposing surfaces which the cut has produced — as equal or as unequal?  If they are unequal, that would imply that ( the lateral surface of ) a cone is composed of many breaks and protrusions like steps.  On the other hand, if they are equal, that would imply that two adjacent intersecting planes are equal, which would mean that the cone, being made up of equal rather than unequal circles, must have the same appearance as a cylinder; which is utterly absurd.


If a thing exists, then either it has magnitude or it does not.  Say it has no magnitude, then if added to another existing thing, it would not make the latter any larger.  That is to say, if something without magnitude is added to another thing, the other thing cannot thereby increase in magnitude.  It follows then, the thing added is nothing.  Thus if something does not lessen the thing that is being subtracted from, and does not increase the thing it is added to, then surely that something is nothing.


If things are many, they must be infinite in number.  For there are always other things between any that exist, and between these, there are always yet others.  Thus things are infinite in number. 




Atoms and Void


Democritus'  greatest contribution was his theory that all is "atoms and void;" the former being completely full and irreducible while the latter, completely empty.  According to him, the universe is made up of atoms flying through emptiness (void) and "hooking around each other."   He said they vary in size and shape.  He reasoned that if a body, an apple say, can be cut with a knife it cannot be completely solid.  Thus, it must consist of many parts with empty spaces between them or the knife couldn't cut the apple.  


Here is a fragment left to us from Democritus on the subject matter:,

A dialogue between the intellect and the senses:


Intellect:  It is by convention that colour exists, by convention sweet, by convention bitter, by convention fragrant, by convention rough or smooth, hot or cold, wet or dry, dark or bright, etc....but in verity, all that exists is atoms and void.


The Senses Retort: Ah, wretched intellect, you get your evidence only as we give it to you, and yet you try to overthrow us.  That overthrow will be your downfall!