Hilbert: Still the Best
Over the last few centuries, the study about the foundations and philosophy of mathematics has led to a diverse spectrum of views. Nonetheless, one of the most remarkable and deepest insights into this subject matter is being held by the most brilliant formalist of all, David Hilbert. While Hilbert’s theory has been widely adopted and studied by many mathematicians and philosophers, it has also been rejected by others. The biggest objection of all is given credit to Gödel and his Incompleteness Theorems. In section 1, we examine the pre-Hilbert formalism and their claims. In section 2, we examine the fundamental problems of the pre-Hilbert formalism. In section 3, we examine more carefully on the details of Hilbert’s formalism. In section 4, we look at the fundamental problems of Hilbert’s formalism. Finally, we examine how Gödel’s Incompleteness Theorems can be used as an argument against Hilbert’s formalism, which is described in section 5. The early formalists claim that mathematics is just a formal game that has stipulations for manipulating mathematical symbols around. Formalists come up with the analogy that best describes their claim as follows: “mathematics is just a game; mathematical objects are
There are some objections to Hilbert’s formalism, big and small. The small objections are described in this section and the bigger objections are described in the following section. The first problem has to do with the fact that Hilbert associates our perceptions and trustworthy reasoning with the finite. The accuracy of our perception on a finite entity is inversely proportional to the amount and complexity of an entity. That means as the entity grows larger and more complex, we tend to lose our grip on a finite entity. For instance, most of us can multiply two small numbers and be certain of the result. However, the chance that we make computational error is relatively high for a product with two large numbers. Thus, “certainty cannot be simply identified with the finite” (Brown, 70). The second problem arises from Hilbert’s claim that the universe is finite. So for some super-large but still finite number, there is no appropriate perceptual experience that allow us to perceive such number, according to Brown. Consequently, large finite numbers “will have to be classified along with transfinite entities as fictions or ideal elements” (Brown, 71). Thus it contradicts Hilbert’s claim that the universe is finite. We have examined some minor objections to Hilbert’s formalism in the previous section. Now, we’ll look at the biggest objections, which is executed by Godel’s First and Second Incompleteness Theorems, and the latter causes more damage than the former. In general, the first theorem shows that formal arithmetic is incomplete. That is, in formal arithmetic, there are statements that are true but unprovable. Since this associates to any theory strong enough to contain arithmetic, it applies to “all of classical mathematics” (Brown, p77). T
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Approximate Word count = 1212
Approximate Pages = 5 (250 words per page double spaced)
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