pages tagged mathsspwhittonhttps://spwhitton.name//tag/maths/spwhittonikiwiki2016-03-25T16:20:13ZRandomly-generated maths paper accepted by a 'journal'https://spwhitton.name//blog/entry/mathgenaccepted/2015-11-18T17:09:12Z2012-10-20T10:14:00Z
<p><a href="http://thatsmathematics.com/blog/archives/102">Mathgen paper accepted! | That’s
Mathematics!</a></p>
Logic proofs resourcehttps://spwhitton.name//blog/entry/metamathproofexplorer/2015-11-18T17:09:12Z2012-06-09T07:45:00Z
<p><a href="http://us.metamath.org/mpegif/mmtheorems.html#mm2b">Metamath Proof
Explorer</a></p>
<p>This is where you can finally find the elusive ¬¬<em>α</em> → <em>α</em> and
<em>α</em> → ¬¬<em>α</em> proofs.</p>
Körner on Platohttps://spwhitton.name//blog/entry/kornerplato/2015-11-18T17:09:12Z2011-10-12T10:49:00Z
<blockquote><p>For Plato an important, perhaps man’s most important, intellectual
task was to distinguish appearance from reality. It is a task required
not only of the contemplative philosopher or scientist but, even more,
of the man of action, in particular the administrator or ruler, who
has to find his bearings in the world of appearance and who must know
what is the case, what can be done, and what ought to be done. To
achieve order, theoretical or practical, in the world of appearances,
which is always changing, we must know the reality, which never
changes. Only in so far as we know that, can we understand and
dominate the world of appearance around us. (S. Körner, <em>The
Philosophy of Mathematics: An Introductory Essay</em> (London: Hutchinson
& Co., 1960), p. 14)</p></blockquote>
Vlastos on Anamnesis in the Menohttps://spwhitton.name//blog/entry/vlastos/2015-11-18T17:09:12Z2011-10-11T08:35:00Z
<blockquote><p>The full-strength doctrine carries not only the implication that
non-empirical knowledge can exist but also, unfortunately, that
empirical knowledge cannot exist. This latter thesis could be
sugar-coated with the plea that since Plato is willing to admit what
we call ‘empirical knowledge’ under the name of ‘true belief’, nothing
is changed except the name. … In refusing the term ‘knowledge’ to
propositions of ordinary experience and of the observational sciences
Plato is downgrading quite deliberately those truth-seeking and
truth-grounding procedures which cannot be assimilated to deductive
reasoning and cannot yield formal certainty; and this has enormous
implications, theoretical, and also practical ones, as can be seen in
the exclusion of disciplines like medicine, biology, and history from
the curriculum of higher learning in the <em>Republic</em>. —G. Vlastos,
‘Anamnesis in the Meno’ in <em>Plato’s Meno in Focus</em> (ed. J.M. Day)
(London: Routledge, 1994), pp. 101–2</p></blockquote>
<p>Fire alarm this morning, I was in the shower, so had to walk across two
quads to the assembly point in just a towel, ouch my feet.</p>
P. Benacerraf, *What numbers could not be*https://spwhitton.name//blog/entry/benacerrafquot/2015-11-18T17:09:12Z2011-09-22T15:27:00Z
<blockquote><p>But there seem to be little to choose among the [possible
set-theoretic models for the natural numbers]. Relative to our
purposes in giving an account of these matters, one will do as well as
another, stylistic preferences aside. There is no way connected with
the reference of number words that will allow us to choose among them,
/for the accounts differ at places where there is no connection
whatever between features of the accounts and our uses of the words in
question/. If all the above is cogent, then there is little to
conclude except that any feature of an account that identified 3 with
a set is a superflous one – and that therefore 3, and its fellow
numbers, could not be sets at all.</p>
<p>[…]</p>
<p>There is another reason to deny that it would be legitimate to use the
reducibility of arithmetic to set theory as a reason to assert that
numbers are really sets after all. Gaisi Takeuti has shown that the
Gödel–von Neumann–Bernays set theory is in a strong sense <em>reducible
to</em> the theory of ordinal numbers less than the least inaccessible
number (1954). No wonder numbers are sets; sets are really (ordinal)
numbers, after all. <em>But now, which is really which?</em></p></blockquote>
Commutativity of ordinary multiplicationhttps://spwhitton.name//blog/entry/multcommutative/2015-11-18T17:09:12Z2011-08-28T09:22:00Z
<p><a href="http://www.dpmms.cam.ac.uk/~wtg10/commutative.html">Commutativity of Multiplication | Timothy
Gowers</a></p>
<p>I found this page interesting and it has me thinking about the best way
to read Maths. Normally I would check every step of every proof on
there, and in doing so would probably have missed the interesting parts
of the page. Instead I went through them pretty quickly. I am often
accused of making too much out of unimportant things in Maths, so
perhaps this is a better approach.</p>
Examiners' reportshttps://spwhitton.name//blog/entry/examreports/2015-11-18T17:09:12Z2011-06-20T15:52:00Z
<p>Examiners’ reports are very amusing. Here’s a paragraph from the 2010
Part A Mathematics:</p>
<blockquote><p>Presentation skills were very mixed. Some candidates wrote to the
standard of professional mathematicians, and their scripts were a
pleasure to read. Others wrote as if grammar and style were far too
costly to be lavished on mere examiners. Many scripts were
unreasonably illiterate. Misuse of the symbol ⇒ to mean ‘then’ rather
than ‘implies that’ was rampant, leading in a number of cases to
assertions that could (and perhaps should) have been marked wrong. Too
often crucial quantifiers, especially ‘there exists’, were missing
(that is, left to the reader to supply).</p></blockquote>
<p>And from 2009:</p>
<blockquote><p>It also gave candidates the opportunity to show that they didn’t know
what “singular” or “nonsingular” meant. Unfortunately, far too many
candidates assumed that if the multiplicity of a root of the
characteristic polynomial is <em>r</em> then there are <em>r</em> linearly
independent eigenvectors. But then, somewhat surprisingly, those
candidates did not deduce in one line that every linear transformation
over ℂ is diagonalisable! Of course, if this should have been the
case, then the problem, which leads to a proof of the Cayley-Hamilton
theorem, would have been trivial (as would the whole of algebra…), but
this didn’t stop those candidates from getting into even deeper water.
…</p>
<p>This question, unwittingly, gave many candidates the opportunity to
show lack of basic understanding. Leaving aside those who define
<em>W</em><sup>⊥</sup> = {<em>w</em> ∈ <em>W</em><sup>⊥</sup> : ……}
or worse … [f]or the next part, the few who realised that some
geometry was going on fared best …</p>
<p>One candidate bravely wrote “in the lectures we were told that [if W
is infinite dimensional, then] <em>W</em><sup>⊥⊥</sup> = <em>W</em>″ –
unfortunately, this is not universally true, for example when
<em>W</em> = <em>V</em> !</p></blockquote>
<p>On a computer screen yesterday I read this from a physics past paper
(paraphrased): “This equation wasn’t done well. Perhaps we should have
provided the auxiliary equation or shown the answer to be worked to? I
didn’t think it was very difficult.”</p>
<p><em>Edit 28/vi/2011:</em> Another someone quoted at me: “it would be an
understatement to say that this question was badly answered: indeed, it
caused utter carnage.” The report goes on for a page about how this
happened.</p>
Constructing the complex numbers from polynomials and ring theoryhttps://spwhitton.name//blog/entry/rings/2015-11-18T17:09:12Z2011-06-13T21:05:00Z
<p>I’ve been putting off revising the <em>Rings & Arithmetic</em> course since I
didn’t follow the lectures or really do the problems back in Michaelmas,
but people keep telling me it’s nice and actually quite easy, so I’m
finally doing some—it’s a compulsory course, but I only have to answer
one question so I thought I might just skip it and try and put more time
into the others. I think that having done some Number Theory helps a
great deal. Fellow Hungarian mathematician who did most of the second
year number theory course at school <img src="https://spwhitton.name//smileys/idea.png" alt="(!)" /> says this helped a lot; they did
this while we were wasting our time with calculus.</p>
<p>So I’m starting to like algebra more—especially now that quotient spaces
don’t scare me quite so much—but this is the usual thing that I really
like Maths when I can do it, but I can’t do it often enough in this
degree course that I stop enjoying it. I wish I’d had a better year so I
could have a better grasp of courses like this; I should have persevered
with Fields, too. Also for any maths undergrads reading, <em>Abstract
Algebra</em> by I.N. Herstein (Chichester: Wiley 1999) is definitely worth
checking out; doesn’t quite do enough for my second year rings course,
but shows you what’s important about the topics.</p>
<p>Here’s the coolest thing I’ve found so far, where we use quotients of
rings to construct ℂ, rather than just saying that there’s an element
<em>i</em> that squares to − 1, which appeals to my constructivist leanings.</p>
<p>We have the ring of polynomials with real coefficients, ℝ[<em>X</em>], no
problem, and now write ⟨<em>X</em><sup>2</sup> + 1⟩ for the multiples of the
polynomial <em>X</em><sup>2</sup> + 1.[1] It’s easy to show that $\langle
X<sup>2</sup>+1\rangle \vartriangleleft \mathbb{R}[X]$. So we can form the
quotient ring ℝ[<em>X</em>]/⟨<em>X</em><sup>2</sup> + 1⟩, and this is essentially ℂ.
So what’s our imaginary unit? Well write
$\mathrm{i} \mathrel{\mathop:}= \overline{X} \mathrel{\mathop:}= X +
\langle X<sup>2</sup>+1\rangle$ and then using the division algorithm we can get
that every element can be written <em>a</em> + <em>b**i</em>: every polynomial may be
written as <em>a</em> + <em>b**X</em> + <em>g</em>(<em>X</em>)(<em>X</em><sup>2</sup> + 1)). And indeed</p>
<p>$$
\mathrm{i}^2 = \overline{X}^2 = \overline{X<sup>2</sup>} = \overline{X<sup>2</sup> + 1 -
1} = \overline{X<sup>2</sup> + 1} - \overline{1} = 0 - 1 = -1
$$</p>
<p>as we require.</p>
<p>[1] The reason we use <em>X</em> rather than <em>x</em> here is to emphasise that
polynomials are <em>not</em> functions, just formal expressions of coefficients
and a few <em>X</em>. <em>x</em> would traditionally denote a variable in a function
rather than just a component of an expression.</p>
The Law of Quadratic Reciprocityhttps://spwhitton.name//blog/entry/quadrep/2015-11-18T17:09:12Z2011-06-02T09:47:00Z
<p>Lecturer and textbooks tell me that this theorem is certainly the most
important theorem in Number Theory and (therefore?) also probably the
most important theorem in all of Mathematics. There are hundreds of
proofs; first one was by Gauss at the age of 19. Lecturer: “if Number
Theory is the queen of Mathematics, then this theorem is its jewel.”</p>
<p>I do not yet fully grasp what the theorem even says (because I haven’t
looked at the proof properly) and I certainly don’t see why it’s so
important yet, but I’m posting it here because I rather like the idea
that the deepest stuff in Maths lies in ℤ.</p>
<p><strong>Definition.</strong> Let <em>p</em> > 2 be prime and let <em>a</em> ∈ ℤ be s.t. $p
\nmid a$; <em>a</em> is a unit mod <em>p</em>. Then <em>a</em> is a <strong>quadratic
residue</strong> of <em>p</em> if ∃<em>x</em> ∈ ℤ s.t. <em>x</em><sup>2</sup> ≡ <em>a</em> (mod <em>p</em>) and
<em>a</em> is a <strong>quadratic non-residue</strong> if not.</p>
<p><strong>Notation.</strong> For any <em>a</em> ∈ ℤ we define the <strong>Legendre symbol</strong>:
$$ \left(\frac{a}{p}\right) \mathrel{\mathop:}=
\left\{\begin{array}{cl} +1 &p\nmid a \text{ and } a \text{ is a
QR of } p, \ -1 &p\nmid a \text{ and } a \text{ is a QNR of } p,
\ 0 &p|a.\end{array}\right. $$</p>
<p><strong>Theorem.</strong> (The Law of Quadratic Reciprocity, Gauss, 1796) <em>For</em>
<em>p</em>, <em>q</em> <em>distinct odd primes</em>,</p>
<p>$$
\left(\frac{p}{q}\right) =
\left(\frac{q}{p}\right)(-1)^{\left(\frac{p-1}{2}\right)\left(\frac{q-1}{2}\right)}.
$$</p>
Bolzano--Weierstrass raphttps://spwhitton.name//blog/entry/bwrap/2016-03-25T16:20:13Z2011-05-25T16:15:00Z
<p>Not sure if this is the proof I am familiar with, but the Viewing Points
proof is my fave. part of first year Pure Maths.</p>
<p><a href="http://www.youtube.com/watch?v=dfO18klwKHg">Bolzano Weierstrass rap (Down with
that)</a></p>