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Contents

  1. Theory of formal systems
  2. Mathematics & Formal Reasoning
  3. What are Formal Methods?

Formal systems can model certain behavior. This is only progress toward understanding formal systems. We soooooo need to go to examples now. When generating theorems you have to follow the rules and never do anything else.

Theory of formal systems

This seems obvious for the MIU-system, but once you perceive the isomorphism in the pq-system that says:. For example, p-p--q is not a theorem, even if you want it to be.


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Is U a theorem of the MIU-system? You could imagine a program generating theorems forever, but no sane human would. Humans would jump right out of the system, easily justifying the metatheorem that all theorems start with M. So how can we tell whether something is a theorem or not? This would seem to require reasoning outside the system. No worries. But one that is worth wondering is whether the decision procedure is mechanical or not.

That is, is there a formal system for reasoning about formal systems? About themselves? Let's try to create theorems that express that certain numbers are prime. It seems easier to start the notion of compositeness. The C-System is an extension of the tq-system. But wait. Is there any reason to believe that we should be able to find typographical rules for non-theorems of a formal system.

Consider the claim that the foundations of mathematics are consistent.


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Suppose not. Then there is an inconsistency in the foundations. We get a contradiction, proving the claim. Bishop was a leading representative of the tendency described as intuitionist or constructivist. This could not be otherwise, since a verificational interpretation of the quantifiers necessarily results in a clash with classical mathematics.

As a matter of presentation, the conflict with classical mathematics had been de-emphasized by Bishop. In Bishopian mathematics, a circle cannot be decomposed into a pair of antipodal sets. A real number a satisfying.

Mathematics & Formal Reasoning

A kinder, gentler constructivism. In Section II. Bishop put such influence to a variety of uses. Bishop narrates a creation story of intuitionism in the form of such an imaginary dialog, where Brouwer completely dominates the exchange.

How to Read Math

We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise Hilbert, , p. I gather that it has met with some degree of success, whether at the expense of giving significantly less meaningful proofs I do not know.

My interest in non-standard analysis is that attempts are being made to introduce it into calculus courses. It is difficult to believe that debasement of meaning. Heyting, who felt that. Bishop anchors his foundational stance in a species of mathematical constructivism in the following terms: To my mind, it is a major defect of our profession that we refuse to distinguish [. This thread will be pursued further in Subsection V. Davis notes that Bishop fails to acknowledge explicitly in his review in the Bulletin that his criticism is motivated by his foundational preoccupation with the law of excluded middle.

Keisler, , p. In short, the constructivist Bishop is criticizing apples for not being oranges: the critic and the criticized are not operating in a common foundational framework. An interesting exchange took place in the pages of the Bulletin and the Notices. One of his goals was to boost lagging sales that were plaguing the publisher at the time, see Halmos, In his autobiography, Halmos described non-standard analysis as a special tool, too special Halmos, , p.

Bishop to speak out against it specifically? We argue in Subsection III. The possible technical reasons as the following: 1. Subsection IV. A related point is that the hyperreal approach incorporates an element of non-constructivity at the basic level of the very number system itself. By replacing Weierstrassian epsilontics40 by a theory of infinitesimals, non-standard analysis thereby removes the bread and butter of constructive analysis, starting with the constructive epsilontic definition of a Cauchy sequence, which involves an explicit rate of convergence, see Bishop, Recall that Brouwer sought to incorporate a theory of the continuum as part of intuitionistic mathematics, by dint of his free choice sequences.

More fundamentally, we would like to argue that non-standard analysis presents a formidable philosophical challenge to constructivism, which may, in fact, have been anticipated by Bishop himself in his foundational speculations, as we explain below.

What are Formal Methods?

There is no reason mathematics should not concern itself with finitely performable abstract operations of other kinds, in the event that such are ever discovered Bishop, , p. Bishop hereby acknowledges that the primacy of the integers is merely what he describes as an empirical fact. The implication is that the said primacy. Perhaps Bishop sensed that a rigorous theory of infinitesimals is both not reducible to finite calculations on the integers, and yet accomodates a finite performance of abstract operations, thereby satisfying his requirements for legitimate mathematics.

Bishop made a foundational commitment to the primacy of the integers a state of mind known as integr-ity, see Subsection VI. In fact, the second-named author has been able to locate a participant in the workshop, who confirmed that Bishop said nothing about non-standard analysis in his oral presentation. I would have pursued the issue vigorously, since I had a particular point of view. I had been considering that question somewhat in my attempts to understand various standards of rigor in mathematics.

The statement would have fired me up Manning, To be sure, additional material is frequently added at galley stage.

The epilogue contains the discussion following the lecture , involving Birkhoff, Mackey, J. Kahane, and others, who have challenged Bishop on a number of points. We now understand why there was no such challenge: Bishop did not say a word about Robinson in his oral presentation. According to such a hypothesis, his objection focused on the use of the extension principle and the transfer principle of non-standard analysis see Appendix A , whose acceptance by the student would necessarily be on the grounds of trust in the teacher.

However, such a hypothesis does not stand up to scrutiny, as we argue below. Davis, V. Komkov, J. Dauben, H. Keisler, D. Tall, and others, and is not a novel interpretation. Thus, M. Physicist V. Komkov notes that Bishop is one of the foremost researchers favoring the constructive approach to mathematical analysis. It is hard for a constructivist. Philosopher of mathematics G. Historian of mathematics J. The usage of the term confirm points to a shared shortcoming.

Note that the term meaning is alluded to elsewhere in his review, as well. The review is evasive as to what exactly Bishop has in mind when he mentions meaning. It is difficult to believe that debasement of meaning could be carried so far Bishop, , p. However, the Crisis text does not contain any other occurrences of the three-word formation.

Notre Dame Mathematical Lectures

Where was debasement of meaning spoken of previously by Bishop? The essay was published posthumously in , but M. Rosenblatt , p. Schizophrenia text was distributed a year or two prior to the Crisis text. Crisis text to refer to non-standard calculus, is in fact a criticism of classical mathematics as a whole. By the time Bishop got around to describing non-standard calculus as a.

Bishop provides an example explaining the meaning of an integer to Brouwer:. Again there is this notion of computability: if the integer is not given directly to be sure that it is finitely computable [ emphasis added— authors]. Toward the end of an imaginary dialog between Brouwer and Hilbert, Bishop has Brouwer express himself as follows on the subject of meaning: For me to do my mathematics in your system would entail a significant loss of meaning54 Bishop, , p. To Brouwer, as interpreted by Bishop, meaning is seen through the lens of notions of computability, numerical meaning, and LEM.

Meanwhile, Hilbert is uninterested in LEM extirpation; for him, the source of meaning lies elsewhere. Our conclusion is that the Bishop-Keisler controversy was, in essence, a foundational controversy, as suggested by J. Dauben, see Dauben, , p. In the midst of the epic tale, Bishop inserts58 a vitriolic paragraph against Robinson-style infinitesimal calculus.

The remarks he added to the galley proofs 5 8 of Bishop, indicate that he was already aware of infinitesimals in the classroom, at least two years prior to writing his book review Bishop, Meanwhile, K. Could it be because Bishop was not interested in field studies?