The King of Infinite Space Read Online Free Page B

common notions are what they seem. They expressassumptions that are more general than his axioms but no less undefended.
    If Euclid’s common assumptions cannot be derived from anything further, they make their claim by means of their inescapability. Without them, Euclid believes, there could be no proof at all. Whatever their inescapability, Euclid’s common notions suggest a question that neither he nor Aristotle ever considered. Can these common notions be faulted because they are incomplete? Whenever an explicit list of common assumptions is offered, after all, it is easy enough to step back and with some assurance point to the assumptions on which the assumptions themselves depend.
    Like any other mathematician, Euclid took a good deal for granted that he never noticed. In order to say anything at all, we must suppose the world stable enough so that some things stay the same, even as other things change. This idea of general stability is self-referential. In order to express what it says, one must assume what it means.
    Euclid expressed himself in Greek; I am writing in English. Neither Euclid’s Greek nor my English says of itself that it is Greek or English. It is hardly helpful to be told that a book is written in English if one must also be told that written in English is written in English. Whatever the language, its identification is a part of the background. This particular background must necessarily remain in theback, any effort to move it forward leading to an infinite regress, assurances requiring assurances in turn.
    These examples suggest what is at work in any attempt to describe once and for all the beliefs “on which all men base their proofs.” It suggests something about the ever-receding landscape of demonstration and so ratifies the fact that even the most impeccable of proofs is an artifact.

Chapter IV
DARKER BY DEFINITION
    Sometimes things may be made darker by definition. I see a cow. I define her , Animal quadrupes ruminans cornutum. Cow is plainer.
    â€”S AMUEL J OHNSON
    T HE ELEMENTS CONTAINS twenty-three definitions. Of these, the first seven, and the twenty-third, are fundamental:
    1.A point is that which has no part.
    2.A line is length without breadth.
    3.The extremities of a line are points.
    4.A straight line is a line which lies evenly with the points on itself.
    5.A surface is that which has length and breadth only.
    6.The extremities of a surface are lines.
    7.A plane surface is a surface which lies evenly with the straight lines on itself.
    23.Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
    Nineteenth- and twentieth-century mathematicians have almost to a man objected to these definitions. Both Moritz Pasch and David Hilbert criticized Euclid because in struggling to say what he meant, Euclid rejected what he knew: things come to an end. If axioms must be accepted without proof, then some terms must be accepted without definition. In his ninth through twenty-second definitions, Euclid is almost impeccable, defining terms that are new by an appeal to terms that are old. A triangle is a figure contained by three straight lines. This is Euclid’s nineteenth definition. Not perfect. What is a figure? But not bad. There remain his initial definitions. A point, Euclid affirms, has no parts. It is the first thing that he says, circumstances suggesting that he meant to say it. And since it is the first thing Euclid says, it is the first definition to which critics object. “This [definition] means little,” Morris Kline argues in Mathematical Thought from Ancient to Modern Times , “for what is the meaning of parts?”
    Yet if Kline intended to rebuke Euclid for a logical mistake, he has done so by making a mistake all his own. The haunch of a cow is one of its parts, but only the word haunch carries meaning. The haunches have