most recent 30 from http://mathoverflow.net 2013-05-19T23:19:00Z http://mathoverflow.net/feeds/user/18805 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102246/sine-and-archimedes-derivation-of-the-area-of-the-circle Marian 2012-07-14T20:05:27Z 2012-07-14T20:05:27Z

<p>The elementary "opposite over hypotenuse" definition of the sine function defines the sine of an angle, not a real number. As discussed in the article "A Circular Argument" [Fred Richman, The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162. Free version <a href="http://mathdl.maa.org/images/cms_upload/0746834216444.di020756.02p0159m.pdf" rel="nofollow">here</a>. Thanks to Aaron Meyerowitz's answer to question <a href="http://mathoverflow.net/questions/72792/who-first-proved-that-the-value-of-c-d-is-independent-of-the-choice-of-circle" rel="nofollow">72792</a> for the reference.], angles might be measured either by the area of a sector of unit radius having the angle or by the arc length of such a sector. If the former convention is adopted then it can be proven using a completely unexceptionable Euclidean argument that $\lim_{x\to 0} \sin(x)/x = 1$. Also, whichever convention is adopted (or so it seems to me), using completely unexceptionable Euclidean arguments, it is possible to prove the angle addition formulas for sine and cosine. Using these two ideas, it is straightforward to find the derivatives of sine and cosine, and from there one can derive an algorithm for computing digits of sine and cosine (and for computing $\pi$) using the relatively sophisticated mean-value version of Taylor's theorem.</p> <p>The equivalence of the two definitions of sine (or of angle measurement) apparently depends on something like Archimedes' postulate: "If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D." (Again, thanks to Aaron Meyerowitz.) Of course, it is just this postulate that Archimedes needed to prove that the area of a circle is equal to the area of a triangle with base the circumference of the circle and height the radius. And something like it is surely necessary to derive any algorithm for computing digits of $\pi$. (Except, and this confuses me a bit, it seems that if we used the area definition of angle, we could derive an algorithm for computing sine without depending on this postulate, and from there we could get an algorithm for computing digits of $\pi$ since $\sin(\pi)=0$.)</p> <p>I am looking in general for elucidation of the conceptual connections between the ideas I have so far discussed and of their background. But here are two more specific questions. First, in what sense is a postulate like Archimedes' needed in the foundations of geometry? (I wonder, in particular, if in a purely formal development we might get by without it, but we would somehow be left without assurance that what we had axiomatized was really geometry.) Also, are more intuitive alternatives to Archimedes' postulate? Second, what is really needed to get an algorithm for computing digits of sine? Does it really require such complicated technology as Taylor series? It seems like if one uses the area definition of angle, one might be able to give an algorithm using unexceptionable Euclidean techniques and without so much as invoking the notion of limit.</p>

http://mathoverflow.net/questions/97008/comprehension-and-ideal-elements Marian 2012-05-15T14:53:29Z 2012-05-15T18:22:40Z

<p>A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations. The abundant ones are the things we get from axioms -- the comprehension scheme in the context of type theory, and set existence principles in the case of set theory. But if you are willing to countenance ideal classes (or relations or whatever), it might seem like just a prejudice to not also countenance ideal elements. Moreover, the history of mathematics is filled with examples of cases where we expanded our ontology by allowing for specific ideal objects like the square root of minus one. (Also, though its connection to my question is hardly clear, forcing is often seen as a means of adding ideal elements.) My general question is: what kinds of systems have been studied that include the idea of postulating ideal elements and/or allow for greater symmetry between the treatment of individuals and classes?</p> <p>My first impression, having thought about the matter not too long, is that combining comprehension principles with principles allowing the postulation of ideal elements tends to lead to inconsistency. For instance, suppose we just have a two-sorted system, with individuals and sets of individuals, and even suppose that comprehension is restricted in the way it is in ZFC. So we have the scheme</p> <p>(1) There is a set such that, for every individual $s$, $s$ is in it iff $s$ is in $X$ and $\phi(s)$.</p> <p>Now add the "inverted" principle</p> <p>(2) There is an individual such that, for every set $Y$, $Y$ contains it iff $Y$ contains $t$ and $\phi(Y)$.</p> <p>Together they lead to a contradiction if we assume that some individual $t$ is in two sets, for by (1) we can form singleton $t$, and by (2) we can find a member of singleton $t$ that is in no set but singleton $t$. Here are three things one might be able to do. First, one might be able to symmetrically restrict both comprehension and inverse comprehension in some way that would result in them not being inconsistent with one another. Second, one might try to build a system using some powerful form of inverse comprehension but without any ordinary comprehension or with only some very weak form of ordinary comprehension. If that could be done, it would show that, although we cannot unrestrictedly have both ideal classes and ideal elements, the choice of which to have is somehow arbitrary. Third, one might might try to build up a model iteratively by alternately applying (forms of) comprehension and inverse comprehension to what one has gotten so far. Have any of these avenues been pursued? Also, are there any natural ways of "inverting" any of the other set existence principles used in set theory?</p>

http://mathoverflow.net/questions/79174/parameters-in-arithmetic-induction-axiom-schemas Marian 2011-10-26T17:20:07Z 2011-10-27T11:31:27Z

<p>The induction schema of Peano Arithmetic is standardly given as the universal closure of $\phi(0)\land \forall x (\phi(x)\rightarrow \phi(x+1)) \rightarrow \forall x\phi(x)$. However, since the language of arithmetic has a name for every standard number, it is not obvious (to a beginner like me) why parameters are necessary in the induction schema; why not restrict to the case where $x$ is the only free variable in $\phi$? Does having parameters in the induction schema really make the system stronger, and, if so, how is that proven? Are there natural theorems that can only or most easily be proven using the stronger system? Is the weaker system of any interest?</p>

http://mathoverflow.net/questions/102246/sine-and-archimedes-derivation-of-the-area-of-the-circle Marian 2012-07-15T00:46:47Z 2012-07-15T00:46:47Z

Yes, you are right, my second comment was a bit overblown. But it seems to me that what you are now saying is quite different from what you were saying in your original comment, so I am confused by the phrase &quot;all I am claiming is that&quot;.

http://mathoverflow.net/questions/102246/sine-and-archimedes-derivation-of-the-area-of-the-circle Marian 2012-07-14T21:26:45Z 2012-07-14T21:26:45Z

By the way, the definition in terms of area seems conceptually more fraught since it presupposes commensurability between areas and linear distances instead of just arc lengths and linear distances. It is only relatively recently in the history of mathematics that people thought it made sense to compare areas to distances.

http://mathoverflow.net/questions/102246/sine-and-archimedes-derivation-of-the-area-of-the-circle Marian 2012-07-14T21:20:59Z 2012-07-14T21:20:59Z

Well, you could also &quot;define&quot; $\pi$ by as the result of some algorithm that outputs its digits. I am supposing that $\pi$ is defined as the ration of the circumference of a circle to its diameter. However, as suggested by my parenthetical comment at the ends of the second paragraph of my question, things might turn out differently if you defined $\pi$ as the area of a circle of unit radius.

http://mathoverflow.net/questions/100440/bounded-operators-and-axiom-of-choice/100444#100444 Marian 2012-06-23T11:46:54Z 2012-06-23T11:46:54Z

This is the standard answer set theorists give to this type of question, and it is useful and interesting. Intuitively, it is a bit unsatisfying since one has the sense that one is never at risk of defining up an unmeasurable set by accident, but it is not always obvious that particular definitions are $\Sigma^1_1$. For that matter, one might even define a set by quantifying over sets of reals, but then there is no guarantee of measurability even given large cardinals, right? Also, just out of curiosity, is there any reason to think sets constructed using DC will be sometimes be measurable?

http://mathoverflow.net/questions/99808/strictly-positive-measures-on-countable-boolean-algebras/99809#99809 Marian 2012-06-18T16:57:09Z 2012-06-18T16:57:09Z

Both of these proofs require choice, but can't you produce a measure on $B$ by induction on $n$ (defining at stage $n$ the value of the measure on the subalgebra generated by $b_1,\ldots,b_n$) in a completely constructive, computable way?

http://mathoverflow.net/questions/97008/comprehension-and-ideal-elements/97018#97018 Marian 2012-05-15T16:52:20Z 2012-05-15T16:52:20Z

It's a good observation that principle (2) is inconsistent with weak pairing, but is that a devastating objection to the idea of doing the second of the three things? The idea of an individual that is in no set is dual to the idea of the empty set. I take the fact that there are only trivial models of (1) together with (2) to be a more serious problem. A worry, though, is that a workable system based on (2) might work by allowing weak pairing and other standard principles on some class of well-behaved individuals and leaving the non-well-behaved individuals as window dressing.

http://mathoverflow.net/questions/96303/pointwise-definable-models/96316#96316 Marian 2012-05-08T10:29:01Z 2012-05-08T10:29:01Z

I think you misread my answer. I do respond to your main question, and the last paragraph of my answer responds to your subsidiary question. About the math tea argument, I guess part of the point of the preprint is that it is not entirely satisfying as it stands. I don't know what you mean by asking whether one definition can define uncountably many sets. The whole point of a definition is that it defines exactly one set. Also, maybe you don't mean to, but by the tone of your remarks, you come across as a bit disrespectful.

http://mathoverflow.net/questions/96303/pointwise-definable-models/96316#96316 Marian 2012-05-08T09:50:02Z 2012-05-08T09:50:02Z

Let me add that the philosophical argument I mention can be slightly strengthened in that if all sets are definable then, using the diagonalization technique, we can informally enumerate the digits of a real number that can't be encoded by a set.

http://mathoverflow.net/questions/79174/parameters-in-arithmetic-induction-axiom-schemas/79257#79257 Marian 2011-10-27T11:54:02Z 2011-10-27T11:54:02Z

That's pretty tricky!