I scored 75. 11/20 on question 6. I used a methoud using indirect symmetries. Works just as well. I had 18 bricks as an answer of course. It is an abstract combinatorial counting problem.
I very much doubt if the person who is tutoring me on M208 really 'owns' the materials or is merely pretending. I suspect the last so a discussion won't work. I haven't got a leg to stand on if I don't score high in the nineties at the exam. Which will be very difficult due to the time constraints anyway.
P.S.
Analysis. The difference between $\mathbf{R}$ and $\mathbf{Q}$ is where mathematics feels more like a creation than an invention. Did mathematics exist before humans populated the earth? Did we discover math or did we create it? This could lead to interesting thought or discussion. Riemann created a function which is continuous but nowhere differentiable. $$f(x)=\begin{cases}\frac{1}{q} \text{ if rational and }x=\frac{p}{q},(p,q)=1\\0 \text{ if irrational}\end{cases}$$
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Ever since reading Kants critique of pure reason I'm more and more convinced that Mathematics is a construction which we use to impose on the reality around us.
ReplyDeleteThe failure of the Frege Russell Hilbert program to reduce mathematics to logic only stengthens my view point.
As you say the real numbers feel more like a construction rather than anything we discover.
Finally as mathematical structures get more and more complex the harder it is to see them directly correponding to reality.
Quantum mechanics with its wavefunction is a case in point. The N body wave function of a quantum system is a function in 3N+1 space dimensions. I find it hard to see this as a real field as some people such as David Bohm would have us believe. As that would mean our world is really a 3N+1 dimensional space rather than the 4 dimensional space time we all love and know. Of course given that most text book problems in quantum mechanics usually only consider 1 particle in a potential then this difference no longer manifests itself and so its easy to miss this subtlety.
I prefer the standard interpretation of Born and Dirac which claims that the so called wavefunction should be seen as a probability amplitude which when I take the modulus squared of it will give me the probabilities associated with quantum events.
On this interpretation the wave function doesn't really collapse. There is no need to invoke signalling between particles at speeds faster than the speed of light to explain correlations between them and so forth.
Yes there is a mapping between the wavefunction and eg the energy levels of a quantum system via the eigenvalues of the Hamiltonian describing the system. But that doesn't mean that the wavefunction is in any sense real
I prefer the
Hi Did you get my comment on the topic of whether or mathematics is primarily a construction or something out there. If you did could you explain why you didn't publish it. If you did not do you want me to send it again
ReplyDeleteBest wishes Chris