Is Non-Euclidean Geometry Favorable to Apriorism?

Non-Euclidean geometry is generally thought to be a “fatal blow” to Kantian transcendental theory. In an earlier article, I expressed doubt about this overly simplistic idea. But recently, as I happened to mull over the question again, an even bolder thought popped into my head — the discovery of non-Euclidean geometry is actually favorable to transcendental theory!

It should be noted that Kant’s transcendental theory is neither empiricism nor rationalism. Kant is not saying that if one first finds a few axioms in human a priori intuition, then one can deduce the true state of the objective world. Kant is answering this question: since human experience of this world is uncertain and unreliable, why can human beings nonetheless obtain such rich and such well-ordered knowledge about this world? Kant’s answer is: the “orderliness” of knowledge does not come from the world itself, but is added to the world by human beings. The world “in itself,” the thing-in-itself, is unknowable; in itself it has no such thing as lawfulness or orderliness, but is instead chaotic and obscure.

What non-Euclidean geometry strikes at is rationalism. Rationalists believe that by means of internally coherent logical deduction, one can arrive at definite truths about this world itself. But non-Euclidean geometry reveals that rational deduction itself is uncertain: several different axiom systems may each be logically consistent, yet there is only one real world. Thus the effort to build the certainty of knowledge on the self-consistency of an axiom system ends in failure.

But Kant did not believe that an axiom system could reveal the essence of the thing-in-itself. The orderliness of the axiom system comes from the orderliness of human thought, and the reason why the theoretical systems constructed by human thought can so successfully “correspond” to the real world, Kant believed, is not that “the subject’s reason corresponds to the world of objects,” but rather that “the object corresponds to the subject.” Objects acquire orderliness through the organization of human thought; this orderliness derives from thought and is added to objects by human beings. It is not something objects possess in themselves.

So, when people look at the world through Euclidean-geometric sunglasses, real space is Euclidean. But is real space in itself really Euclidean? We later discover that when people look at this world through non-Euclidean geometry, the real world can likewise be understood as non-Euclidean. General relativity has not proved that real space can only be non-Euclidean. As Poincaré pointed out, we can construct a Euclidean model in which the lengths of objects are compressed under the action of gravity; this model can be mathematically completely equivalent to a non-Euclidean model, except that the non-Euclidean model is merely more convenient to calculate with in certain situations. That is to say, the question of whether the real structure of space is Euclidean or non-Euclidean is probably one that can never be answered, and the transcendentalist view is that the orderliness of the world is conferred by human thought; neither Euclidean nor non-Euclidean geometry is a direct description of the world itself. In this way, non-Euclidean geometry is favorable to transcendental theory.

In Poincaré, transcendental theory was transformed into conventionalism. Conventionalism and transcendental theory are extremely similar. But conventionalism evades the “search for certainty” in science, contains the germ of a rejection of objectivism, and is even more inevitably bound to move toward relativism. Although personally I may perhaps lean more toward conventionalism, I feel that if one wants to preserve many of the merits of conventionalism and instrumentalism while avoiding relativism and anti-objectivism, and also continue the search for certainty, then returning to transcendental theory may be a pretty good choice.

January 16, 2007, 21:19

at Yangroupao Residence

Latest Comments
  
luxin

2007-03-18 14:49:41 [Reply]

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I completely don’t understand it 
But the metric properties of space are still related in some way to space itself; that is to say, for the universe, there may not necessarily be a metric that satisfies Euclidean properties

  
Gu Wu

2007-03-18 17:49:10 [Reply]

Luxin ah? 
As for the significance of non-Euclidean geometry for transcendental theory, I haven’t really figured it out. Now, thinking back on it, I tend to believe that it is neither a fatal blow nor especially favorable.  
If one wants to make the universe satisfy a metric with Euclidean properties, I’m not quite sure how; but I believe that through modifying a series of basic laws such as the law of gravity, etc., achieving mathematical equivalence is still possible. In any case, the key issue is: what exactly is the so-called “space itself”? We can talk about how space is and how it is not, but all such talk is inseparable from mathematical and physical theoretical construction. We always use human language to describe “space itself,” so why is our description reliable and credible? Why has science been able to succeed so remarkably? What modern philosophers have devoted themselves to answering is precisely this question. From Kant to Husserl, they all devoted themselves to establishing the foundations that make science possible. Of course, even without the efforts of philosophers, science would still develop at a rapid pace; but philosophers like to get to the bottom of things, to ask after the ultimate source of knowledge, and Kant’s transcendental theory is precisely such an effort.

Translated from the Chinese original with AI assistance. The original text is authoritative.