By K.O., H.N. Shapiro, et al. Friedrichs

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**Extra resources for Integration of Functionals. Seminar**

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But instead go and check briefly, every 25 seconds or so, whether the water has started boiling. Time yourself too. Here’s why it works. It does not matter whether the measuring apparatus is made up of a laser, or, say, a set of lasers with processing computer, or all of the above including a human observer - say, the operator of the apparatus - as long as human consciousness - the human brain - also has a wave function. In this way the measurement comes from 28 Wave Functions: Mathematics Is Reality the establishment of a relationship between the wave functions of the measuring apparatus and the measured phenomenon.

Heisenberg’s Uncertainty Principle (aka Indeterminacy) 37 The French astronomer and mathematician Pierre Simon Laplace (born 1827), famously once said that if we were to know all of the universe’s atoms and constituents’ exact locations and exact momentums at any particular point in time, we would then in principle be able to compute and predict with infinite accuracy the state of the universe at any arbitrary point in time henceforth - if we only had the requisite computing power at our disposal.

The set of rational numbers is seemingly much larger than the set of natural numbers, because, whereas natural numbers contain no numbers between any two numbers (there is no natural number between 1 and 2, for instance), rational numbers do contain an infinity of numbers between any two natural numbers. Yet natural numbers and rational numbers can be matched one-to-one, and no leftover infinity of remainders remains. 3. e. g. e. the number of times they crop up among the other natural numbers - becomes smaller and smaller even as the numbers become bigger: they progressively become scarcer - fewer and farther between (23).

### Integration of Functionals. Seminar by K.O., H.N. Shapiro, et al. Friedrichs

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