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Daniel Boorstin’s “The Discoverers” begins with the discovery of time; a choice that is as fundamental as it is unlikely for most people. The choice of the concept of time as a discovery reflects a way of thinking, distinct from a formulaic test oriented focus geared toward definitive answers or definitive subjective response. It should be required reading in high school.

But how real is time? This is the subject of Palle Yourgrau’s “A World Without Time”. Part biography of Kurt Gödel, the logician, and Albert Einstein, the physicist, it explores mathematic philosophy in the context of special and general relativity during the second half of the 20th Century. The heart of the book is Gödel’s argument that Einstein’s general relativity does not turn back time on itself, but by doing so, makes time non-existent in that universe. Mr. Yourgrau has made an academic career out of the resurrection of Gödel, who he believes is misunderstood and underappreciated. As a philosopher, Mr. Yourgrau’s book is not an easy read, although it was purportedly dumbed down for the uninitiated. My math and science background is not particularly strong, so the fault partly lies with me. His usage of terms of art instead of common English with examples, made this a difficult read. For instance, in describing the mathematical formalism of David Hilbert (mathematical intuition is replaced by axiomatic proofs based on theorems), he uses semantics and syntax before explaining what these mean in logic. For those with a background in logic or computer science this will not present an issue, but if you don’t know that semantics relates to interpretation (i.e., what something “means”) and syntax to the structure or rules of construct of the system, you are lost. For Gödel, formal proofs could not capture all that was true about numbers or mathematics. The inherent assumptions and limitations make mathematical proofs self-limiting.Relationships may be proven to be consistent (not contradictory), but within that system will never be complete (establish all truths about the subject). Formalism (deduction) is a process of elimination, that at best gets you close. In fairness, mathematical intuition may precede mathematical proof, which takes more time to develop.

While recognizing the incompleteness of mathematics, Gödel like Einstein, believed that mathematical objects and properties exist objectively and independently of knowledge of them by the human mind. This may be an overstatement of their belief, as alternatively it may reflect their epistemological desire to learn what may be knowable, and not to accept Kantian doctrine of unknowability and to ontologically limit investigation to know what is. They both accepted mathematic intuition refusing to substitute syntax (proof) for semantics (truth). Gödel’s incompleteness theory (all truths could not be known through formal proofs) limits proven relationships discerned through mathematical systems (dependent on human creation or discovery). Mathematical perception is unproven and incomplete. It is unclear to me how mathematics is not limited (at least to us) by the biology of the human mind and thus is objective and independent. If recognition of a fudge factor-incompleteness- makes it so, mathematics is a limited truth. If it may be objective and indepedent in expressing relationships in the actual or relative world, it may be so, or wishful thinking. The premise is that unknown mathematical objects that exist objectively and independently will be discerned by us or another more intelligent being or mechanism once we stop looking through a glass darkly. The long and the short of the extended argument is that mathematics is a tool.

The strength of the book is its discussion of mathematic theory independent of and as it relates to the concept of time. Gödel and Einstein were walking companions while they both were at Princeton’s Institute of Advanced Study, a substitute home during WWII for those who formerly participated in the Vienna Circle of intellectuals. Gödel devised limited cases such that mathematical intuition could be effectively ruled out. To develop his incompleteness theorem he used a formula that was provably unprovable, but intuitively true (i.e., consistent but incomplete). In the case of general relativity he develops a theoretical universe that comports with physical laws in the actual world but is a closed system in which time travel (going effectively backwards in time) is possible. For Gödel if what has passed has never passed and the past and the future are the same, then time is an illusion. Perhaps time in Einstein’s geometric space-time is not a fourth dimension, since it cannot exist in every universe.

There is an interesting AT&T advertisement where an adult asks kids what is bigger than infinity. One answers infinity plus one. The other tops this and answers infinity times infinity. Georg Cantor tried to find the size of infinity through his continuum hypothesis. He ended up in a sanatorium as did Gödel.
Within a particular system the size of infinity may be unknowable, but would the kids be wrong in having a multiple of infinite systems. Does infinity need to be a straight line (or space) presumed to be moving forward from a point, or can it be closed system(s) (circular, cylindrical, figure eights, etc.) where in a relativistic world “time” overlaps in a continuous Ground Hog Day. Could those infinite systems overlap, be concentric, parallel, etc. ?

I suspect that there are better books to explore these topic than “A World Without Time”. If you have some grounding in the subject, you likely already know the content save for the cursory history. I am looking forward to reading some real fiction. My mind hurts.