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**1 - 6**of**6**### Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits

"... The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicia ..."

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The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid the foundation for Gödel's work, describes the unique method used by Gödel to prove his famous incompleteness theorem, and discusses the farreaching mathematical implications thereof. 2 I.

### CHAPTER 3 MATHEMATICS AS A BIOLOGICAL PROCESS

"... I've been learning some biology at this meeting, but what can I contribute to the discussion? The answer is, not much. I do discrete mathematics, in fact, metamathematics. In my work I use a notion of complexity, and it would be great if this complexity concept had something to do with biology. Unfo ..."

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I've been learning some biology at this meeting, but what can I contribute to the discussion? The answer is, not much. I do discrete mathematics, in fact, metamathematics. In my work I use a notion of complexity, and it would be great if this complexity concept had something to do with biology. Unfortunately it doesn't. Biology deals with very complicated systems, but I don't think my work on complexity can be applied to biology. Instead I've taken the idea of complexity from biology and I'm using it to try to understand what mathematics can and cannot achieve. That's called metamathematics. So in this talk you are going to see an idea from biology applied in an unexpected way. My story begins about a century ago with David Hilbert, a famous German mathematician, who believed that mathematical truth is absolutely rigorous, completely black or white, never gray. And he wanted to demonstrate that in the most clear-cut, straightforward possible

### Gödel’s Incompleteness Theorems

"... In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epoch-making paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s ..."

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In 1931, when he was only 25 years of age, the great Austrian logician Kurt Gödel (1906– 1978) published an epoch-making paper [16] (for an English translation see [8, pp. 5–38]), in which he proved that an effectively definable consistent mathematical theory which is strong enough to prove Peano’s postulates of elementary arithmetic cannot prove its own

### Zeps, D. Mathematics is Physics 1 Mathematics is Physics

"... In series of articles we continue to advance idea that mathematics and physics is the same. We bring forward two basic assumptions as principles. First is the primacy of life as opposed to dominating reductionism, and second – immaturity of epistemology. Second principle says that we have reached st ..."

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In series of articles we continue to advance idea that mathematics and physics is the same. We bring forward two basic assumptions as principles. First is the primacy of life as opposed to dominating reductionism, and second – immaturity of epistemology. Second principle says that we have reached stage of epistemology where we have stepped outside simple perceptibility only on level of individuality (since Aristotle) but not on level of collective mind. The last stage have reached only most of religious teachings but not physical science that is still under oppressive influence of reductionism. This causes that what we call research in physical science turns out to be simply instrumental improvement of perception within visional confinement we call field of information. We discuss and try to apply principle that within field of information we can’t invent or discover anything that doesn’t existing.

### EXTENDING CANTOR’S PARADOX A CRITIQUE OF INFINITY AND SELFREFERENCE

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"... Abstract. This paper examines infinity and self-reference from a critique perspective. Starting from an extension of Cantor Paradox that suggests the inconsistency of the actual infinite, the paper makes a short review of the controversial history of infinity and suggests several indicators of its i ..."

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Abstract. This paper examines infinity and self-reference from a critique perspective. Starting from an extension of Cantor Paradox that suggests the inconsistency of the actual infinite, the paper makes a short review of the controversial history of infinity and suggests several indicators of its inconsistency. Semantic self-reference is also examined from the same critique perspective by comparing it with self-referent sets. The platonic scenario of infinity and selfreference is finally criticized from a biological and neurobiological perspective. 1.

### Why Not Philosophy? Problematizing the Philosophy of Mathematics in a Time of Curriculum Reform

"... and others, the author draws parallels between social constructivism and a humanism philosophy of mathematics. While practicing mathematicians may be entrenched in a traditional, Platonic philosophy of mathematics, and mathematics education researchers have embraced the fallibilist, humanist philoso ..."

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and others, the author draws parallels between social constructivism and a humanism philosophy of mathematics. While practicing mathematicians may be entrenched in a traditional, Platonic philosophy of mathematics, and mathematics education researchers have embraced the fallibilist, humanist philosophy of mathematics (Sfard, 1998), the teachers of school mathematics are caught somewhere in the middle. Mathematics teachers too often hold true to the traditional view of mathematics as an absolute truth independent of human subjectivity. At the same time, they are pushed to teach mathematics as a social construction, an activity that makes sense only through its usefulness. Given these dichotomous views of mathematics, without an explicit conversation about and exploration of the philosophy of mathematics, reform in the teaching and learning of mathematics may be certain to fail. The teaching and learning of mathematics is going through tremendous changes. The National