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Writing in Mathematics Seminar Video

ACT@UCR Seminar: Mathematics in the 21st Century - John Baez

Writing in Mathematics Seminar - what

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This rule is called the law of detachment. Though it may seem innocuous and - by design - intuitive, its foundational role in logic spells the doom of all certainty of knowledge. But before we can explore how detachment impacts scientific truth, we need to understand its place in logic. Implications and the law of detachment are the go here upon which mathematical truths, or theoremsare proved. For instance, consider the following number-theoretic statements:. Suppose we already know that statements A Writing in Mathematics Seminar B are theorems - in other words, they are true - and we want to prove that C is a theorem. If we show that D is true, then intuitively, we Writing in Mathematics Seminar apply the law of detachment to A and D to conclude that C is true. Sure enough, showing D is true is as simple as substituting 3 in for x in statement B, as we know B is true for any number.

Simple, no? The steps of logic we used are all perfectly sound. We would have needed additional proofs to show those two statements were true, allowing us to apply detachment in the above proof.

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Those additional proofs would have themselves required assumptions upon which to conclude that A and B are theorems. Where does the sequence of assumptions end? Link ends with axioms.

Somewhere in our line of logic, there must be a statement which we declare to be a theorem without proof. Such a statement is an axiom. An axiom "kicks off" a chain of detachments as an initial antecedent, allowing us Writing in Mathematics Seminar conclude a chain of consequents. It will be useful to invent nomenclature for two different types of axioms:.

An absolute axiom is a statement assumed to be true because it serves as the foundation for a particular area of logic, or because it is considered self-evidently true. A quasi-axiom is not an axiom in the conventional sense.

Instead, it is a statement assumed to be true within a particular context, with the understanding that proof either A exists, or B ought to be laid out, to verify the assumption. The selection of both absolute and quasi-axioms are arbitrary for any given situation, and the line between the two categories may move freely depending on context. Mathemayics

This is outside the scope of our current discussion, but if you consider in the remainder of what we discuss that our methods of deduction are not certainly correct, the conclusions become even more damning. There is a striking correspondence between fields of mathematics and disciplines of science, and their usage of axioms, implications, and detachment.

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We often regard the scientific method as a strategy for finding truths Writing in Mathematics Seminar the natural world, but this is not quite here case. Here, I argue that scientific inquiry does not provide evidence for conclusions ; it provides evidence for implications. Consider an experiment where a chemist combines hydrochloric acid HCl and sodium hydroxide NaOHseeking to identify the products of the reaction. She ultimately identifies the products to be sodium chloride NaCl and water H Seinar O.

What can we deduce from this experiment?

There were countless assumptions made during the experiment which deserve consideration. Some such quasi-axioms include:. These quasi-axioms specifically concern the performed experiment, but there are even larger, more fundamental assumptions at play too. To perform her test and analysis, the chemist relied Mathemxtics on the core understood principles of chemistry, such as our Writing in Mathematics Seminar models of atoms, molecules, reactions, and so on. Every experimental outcome depends on infinitely many small and large assumptions, some intentional and some inadvertent. And every experiment provides evidence for the implication from these assumptions to a particular conclusion, not evidence for the conclusion itself.

To prove the conclusion, you need the law of detachment, which requires proving - or, more commonly, blindly accepting - that the antecedents are true.]