Given that Euclidean geometry still works well enough to describe everyday life, you might wonder whether it makes any difference that there are now other geometries, some of which work better than Euclidean geometry in some situations. In fact the consequences have been profound.
Non-Euclidean geometry marked the end of an entire line of human thought, one that had dominated intellectual efforts in the West for centuries. No longer did thinkers search their divine intuitions to find self-evident truths, then use those truths as the bricks of an unquestionable edifice. The assumptions of all subsequent systems of thought would have to be provisional and based on experience. And when the assumptions are only approximately right, the conclusions drawn from them may be even less correct. Consequently, intuition can never be turned off. If you intend to apply your ideas to the real world, then each conclusion, no matter how rationally it follows from its premises, must be checked for plausibility.
In mathematics, non-Euclidean geometry began a century-long inspection of the foundations of mathematics, which resulted in a complete reinterpretation of its nature and its applicability to the real world. In his book The Non-Euclidean Revolution, Richard Trudeau put it this way: "Formerly the essence of a mathematical system was considered to lie in the combination of its logical skeleton and the meaning attached thereto; the modern view is that a mathematical system is, at root, only a logical skeleton, to which meaning may or may not be attached." The system’s fundamental terms – in geometry, terms like point, line, and plane – are left undefined. The axioms of the system are not asserted to be true, they are simply assumptions. The theorems are not asserted to be true either; they are simply what follows when you make these assumptions about these undefined terms. And no one guarantees that the application of any particular mathematical system to the real world will yield any true or useful results.
Morris Kline wrote: “By 1900 mathematics had broken away from reality; it had clearly and irretrievably lost its claim to the truth about nature, and had become the pursuit of necessary consequences of arbitrary axioms about meaningless things.”
And so, reason has ceased to be coercive in any practical sense. The participation of the reader is now required. The reader assigns the undefined terms meaning out of his or her own experience – there is no other source of meaning in the system. It is the reader, then, who asserts the truth of the assumptions based on the meaning he or she has assigned to the undefined terms. The system as a whole has no meaning or truth until the reader assigns it some.
In the last two centuries, reason and common sense have suffered one blow after another. Non-Euclidean geometry was the first of many. In physics, Einstein's relativity contradicted our intuitions about time and space, while quantum mechanics created an image of the physical world which our imaginations have yet to assimilate. Biologically, many of the comforting assumptions we had made about our species have been destroyed by Darwin's theory of evolution. Freud’s psychology put forward the idea that our own minds are not what we thought they were, but rather we are motivated by unconscious desires. Behaviorism went so far as to deny the existence of the conscious mind altogether. Marx asserted that the beliefs of a society are formed by power, not by reason; the ruling ideology, he claimed, was simply the ideology of the ruling class. Social scientists have undercut the Enlightenment notion of progress, as well as the foundations of our moral values. In philosophy, the concept of truth itself is now suspect.
I would go too far if I claimed that non-Euclidean geometry was the direct cause of all these events. Most of the prime movers in these areas probably never thought about non-Euclidean geometry one way or the other. And yet there is a causal relationship. When a dam breaks and the town below is flooded, the townspeople will say that the failure of the dam caused the flood. But the water’s point of view is quite different. The flood waters know nothing about the dam, they just flow to the low spot.
Euclidean geometry was precisely such a dam, protecting the worldview of the Enlightenment. Any 18th century thinker who questioned to the ability of human intuition to find truth, or the usefulness of pure reason, ran directly into the authority of Euclidean geometry. At the time of Jefferson, it was undeniable that pure reason could lead to results of absolute certainty and great practical usefulness. No one could deny the existence of such truth, when so much of it was sitting right in front of them.
In our day the dam has burst and nothing has replaced it. You may doubt what you will about reason, intuition, common sense, or any other human faculty without anyone mentioning geometry to you, or pointing you in the direction of any similarly solid edifice. To read Descartes or Spinoza or even Jefferson is to see into another world, a world that seems simpler, safer, and somewhat quaint. There is, however, no going back.
Return to the outline of The Unreasonable Influence of Geometry