• Korean Journal of Cognitive Science
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• v.6 no.3
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• pp.5-23
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• 1995

According to a well-known argument (so-called the Godelian argument) proposed by Lucas. Godel's theorem refutes the thesis of mechanism. that is, the thesis that human cognitive system is no more than a Turing machine. The main aim of this paper is to show that this argument is not successful. However. I also argue that many pre-existing objections (by Benacerraf, Slezak. Boyer. Hofstadter etc.) to Gooelian argument are not satisfactory. either. Using Tarski's theorem. I then strengthen what I caII the consistency objection to Godelian argument. In my dilemma objection obtained. Godelian argument doesn't work because the argument has a false premise if we have the concept of global truth and the argument cannot be stated if not.

• Journal for History of Mathematics
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• v.1 no.1
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• pp.71-75
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• 1984

Whether the complete Hilbert program could be carried out was rendered very doubtful by results due to Godel. These results may be roughly characterized as a demonstration that, in any system broad enough to contain all the formulas of a formalized elementary number theory, there exist formulas that neither can be proved nor disproved within the system. In this paper, Godel's incompleteness theorem is explained roughly moreover formul system and machines being refered, related to his theory.

• East Asian mathematical journal
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• v.24 no.5
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• pp.465-476
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• 2008

Even though Godel's theorem is remarkable to mathematics teachers, it is not simple to understand the proof in detail. It would be useful for us to understand the basic ideas and the proving process of the proof. In this note, we suggest a proposal for the purpose.

• Journal for History of Mathematics
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• v.14 no.1
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• pp.47-60
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• 2001

The present study develops the given theme “Mathematics and Reality” along two lines. First, we explore the answers, in its various facets, to the following question: How is it possible that mathematics shows such wondrous efficiency when explaining nature\ulcorner In addition to a comparative analysis between empiricism and rationalism, constructivism as a function of idealism is compared with realism within the frame provided by rationalism. The second step involves limiting our discussion to realism. We attempt to explain the various stages of mathematical realism and their points of difficulty. Postulate of parallels, Godel's theorem, continuum hypothesis and choice axiom are typical examples used in demonstrating undecidable propositions. They clearly show that it is necessary to mitigate the mathematical realism which depends on bivalent logic based on an objective exterior world. Lowenheim-Skolem theorem, which states that reality is composed not of one block but rather of diverse domains, also reinforces this line of thought. As we can see the existence of undecidable propositions requires limiting the use of reductio ad absurdum proof which depends on the concept of excluded middle. Consequently, it becomes obvious that bivalent logic must inevitably cede to a trivalent logic since there are three values involved: true, false, and undecidable.

• Journal for History of Mathematics
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• v.29 no.1
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• pp.31-44
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• 2016

This essay aims to suggest roughly the "philosophy of limits." The limits mainly refer to those of human experiences and rational thoughts. The philosophy of limits consist of three theses and two consequences(L, M). (1) The limits are necessarily supervenient in the course of searching knowledge. (2) The limits cannot be dissipated ultimately. (3) To recognize the limits is not only an intellectual recognition but also a beginning of whole personality's reaction. (L) It is a rational decision to accept the limits and leave the margins (yeoback/yeoheuck) rather than to try to remove them. (M) To leave the margins (yeoback/yeoheuck) is characteristic of being human, and enables one to harmoniously communicate with others. To justify the philosophy of limits, this essay examine the limits discussed in mathematics and philosophy: set theory, Godel's Incompleteness Theorem, Galois Theorem in mathematics; and Hume, Kant, Kierkegaard, and Wittgenstein in philosophy. I try to interpret consciousness of limits in various cultures. I claim that consciousness of the limits contribute to lucidity of human identity, communication between persons, stimulation of creative thinking.

• Journal for History of Mathematics
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• v.17 no.4
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• pp.27-36
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• 2004

This paper addresses G$\ddot{o}$del's critique of Turing's mechanism that a configuration of the Turing machine corresponds to each state of human mind. The first part gives a quick overview of Turing's analysis of cognition as computation and its variants. In the following part, we describe the concept of Turing machines, and the third part explains the computational limitations of Turing machines as a cognitive system. The fourth part demonstrates that Godel did not agree with Turing's argument, sometimes referred to as mechanism. Finally, we discuss an oracle Turing machine and its implications.