Kurt Gödel was a mathematician born in Austria in 1906. The Gödel family made their money in textiles, but Kurt’s father was not a well-educated man. His mother, on the other hand, had undergone formal schooling, and instilled a firm belief in getting a good education in Kurt. As a result, he completed his studies at the top of his class in high school, and then went on to earn several degrees from the University of Vienna, including his doctorate in mathematics. What is so ironic about Gödel’s life is that, though he spent nearly his entire life studying theories of logic, he was a hypochondriac who feared being poisoned. He died from a lack of nutrition, and starvation, because he was convinced that someone was trying to kill him by putting something harmful in his food.
His most famous work, still discussed today, is the Incompleteness Theorem in mathematics, which consists of two parts. One of the main themes of his work suggests that “the axiom system must be incomplete,” and that not everything can be sufficiently proven when it comes to the axiomatic mathematical system (Devlin 2002). Originally written in 1931, his theorems have caused much controversy about what math and logic should truly mean, since many theorists believe in the absolute truths and outcomes that math has to offer. Gödel challenged this vein of thought, and created the belief that there might be more than one correct answer when it comes to problem solutions.
What Gödel’s theorem seems to do is it “imposes some [sort] of profound limitation on knowledge, science, mathematics” (Gödel’s Theorem 2007). It takes the concept of axioms, which are indisputable truths, and places a certain level of questioning upon them, so that it sort of breaks apart logical answers and conclusions that we may come to when figuring out a problem. This can be incredibly dangerous on the surface, because it may (and probably has) opened up an indefinite number of solutions to problems. In doing so, there is no distinct way to prove that something is correct, so then the sciences we should see as definitive, become a bit more subjective. Gödel’s critics feel that this type of thinking throws everything off balance, and can interfere with other scientific principles.
The first part of Gödel’s theory seriously questions the usage of proofs in mathematics, which specifically affects the area of geometry. Thus, for every proven mathematical statement, another one can be conversely constructed that is not necessarily provable. They may be implied by the set of axioms, because they are able to be constructed, given the conditions of the axioms. But, all the same, that does not mean that they should be constructed, because, in turn, they may end up contradicting themselves. What is accepted as truth in math is not necessarily proof. The two terms are not interchangeable according to Gödel, which would lead us to prove concepts that are not necessarily valid. This might seem like a waste of time, but the best test of something’s validity might be to in fact explore other facets of an argument in order to eliminate any shade of doubt.
The second part of this incompleteness theory involves consistency for provable theorems. It suggests that, somewhere in the many linear equations to be solved, there is something that can eventually break off, and not be in line with the rest of the proof that these mathematical concepts are in fact true. When defining natural numbers, this actually defies logic, because Gödel states that a formal system which aims to do so can specifically and definitively prove these numbers. Somewhere in that number system will be some statement or axiom that will be neither true nor false. And thus, since it cannot be proven, does not make it decidedly so.
As with any theory, there are limitations on Gödel’s incompleteness theorem that allow for some debate. For example, just because one is questioning whether or not something is true, does not mean that all cases call for such. As an illustration, it could be seen in terms of the Ancient Greek Liar Paradox: “a person stands up and says ‘I am lying’; if the person is lying, then the statement is true, so they are not lying; and if they are not lying, the statement is false, so they are lying” (Devlin 2002). So, either way, it is neither true nor false, and thus not worth the effort in saying it. That is how many felt about Gödel’s theorem.
But, in any event, he did leave quite a mark on the world of mathematics because he chose to refute the status quo, and not simply accept proven mathematical concepts as true, because there are very few absolute truths, even when it comes to science.
References
Devlin, K. (2002). Kurt Godel—separating truth from proof in mathematics. Science,
298(5600), 1899-1901. Retrieved November 22, 2007 from Academic Search
Premier.
“Godel’s Theorem.” (2007). Center for the Study of Complex Systems, University of
Michigan. Retrieved November 22, 2007 from
http://www.cscs.umich.edu/~crshalizi/notebooks/godels-theorem.html.
Charles Carlini is the founder of www.SimplyCharly.com, an educational suite of websites for students and teachers that brings to life, in a compelling and engaging manner, some of the world's most prominent historical figures.
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