Example: A line of some particular length can (in theory at least) be divided into an infinite number of segments. A line twice as long can likewise be cut into an infinite number of segments. So the second line's infinite segments would be more than the first line's infinite segments.

The number of all integers is infinite. The number of all prime integers is likewise infinite, but it is smaller than the number of all integers. The number of all fractions between the integers 1 an 2 is infinite, but it is a very small infinity.

The rational numbers form a countable infinite set but the set of reals is uncountably infinite. There's various demonstrations of this on the web.

of quantum mechanics (the many worlds alternative). Everything is just probabilities until measurement takes place
upon which there appears to be a discontinuous jump to a definite observable (if that makes sense to you). There may be an infinite number of universes each with a different outcome to the measurement or observation (where some scientist say consciousness is needed for this to occur).
Also there is the cosmological theory of different universes sprouting continuously where the universe we are in is compatible with consciousness.

and should not be taken seriously as a scientific thesis unless it produces a useful technique for calculating experimental outcomes and predicting previously unknown phenomena

Quantum mechanics can be understood as an attempt to grapple with the fact that the world appears to involve tiny discrete items; and since there is a limit to the scale of the items we can actually produce, there is also a limit to the scale of the disturbances we can create and hence a limit to the scale of the objects we can probe

The world may exist at smaller scales than we can probe but what happens there is beyond our reach

One could take the view that no such smaller scales exist for us, and that all the mysteries of the small world appear simply because of some mysterious "reduction of the wave-function" that occurs at the moment of our measurements, somehow selecting this world from a large number of "possible worlds" -- but that seems to me a negative philosophical hypothesis, which closes off the possibility future development

Perhaps a more productive avenue of approach is to attempt to discover "book-keeping rules" which enable us to model existing phenomena on the basis of hypothetical objects: "quarks" seem to be an example of such "book-keeping rules" for "objects" that are unobservable in principle but that appear to bring a more systematic order to the observable world

Notice that it is incoherent, as a matter of natural philosophy, to leap from a "possible worlds" interpretation of quantum mechanics to a grandiose "possible worlds" view of our cosmos: the first problem confronting the founders of quantum mechanics was that the world of the very small appeared to be governed unavoidably by strange statistics, but that this statistical weirdness usually disappeared as one moved towards larger classical objects. The solution to this problem seems to be something like "the law of large numbers" in the context of statistical mechanics -- so any philosophical justification for "possible worlds" interpretations vanishes for objects on a classical scale. Why then would anyone expect to see it again at cosmological scales and think the statistics might be creating an unimaginable multitude of "universes"? It is an unsupportable speculative leap, which happily skips over much more mundane and difficult problems (such as attempting to unify parts of quantum mechanics with relativity)

The jump from the wave-function to an observable is also referred to as the collapse of the wave function (you probably are aware). You're probably familiar with Schrodinger's cat.
And yes the many worlds interpretation is a philosophical one.

in a (not very good) textbook on quantum chemistry, about forty years ago: the author jabbered for several pages on the idea, and so (being naive) I took it seriously for a while and tried to understand what could be done with it

And despite spending quite a while on it, I got absolutely nothing from it

Planck's path to his constant made sense; Einstein's explanation of the photoelectric effect made sense; the Bohr atom made sense (even though it is quite limited in application); how deBroglie came up with his wavelength made sense; how Schrodinger found his wave equation (using the deBroglie wavelength) made sense

I find nothing wrong with probabilistic thinking in physics. Putting such methods on a firm foundation was an imperishable achievement of Boltzmann. And (if my memory serves) Einstein first suggested that a probabilistic interpretation would reconcile the electromagnetic wave equation with the "photon" notion

But "reduction of the wave function" has never helped me understand anything: it has remained a riddle, wrapped in a mystery, inside an enigma

I have read Schrodinger's well-known cat essay. He stated the issue clearly enough, but I suspect his object was really to point out a developing conceptual difficulty at the foundation of physics, rather than to advocate a definite point of view

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At some level, science will always be philosophical. That level is the one which goes beyond our current knowledge. Things involving both universe-scale and subatomic scale may always be addressed philosophically to some degree. We can't observe at either level, so we see unclearly. That's the realm of philosophy, really.

To wit: I have a school textbook printed in the 1870s. Its title is "Natural Philosopy." It's a basic science book for high school aged children, and covers all of the sciences. It's very interesting to read, since it exposes how little we knew, even at that late date, historically. The Theory of Evolution was just beginning to be talked about in such textbooks, and was far from being accepted.

We knew nothing about plate tectonics, and could look no closer than optical microscopes and no farther than telescopes could reveal. I really recommend reading a 19th century Natural Philosophy book for anyone interested in science. It's an eye-opener, and will make you appreciate how much more we know now than we did then.

Newton's Mathematical Principles of Natural Philosophy is without question one of the great classics of what we now call "science"

What distinguishes that work as "science" (rather than "philosophy" ) is Newton's dedication to "saving the phenomena," as he devises his hypotheses and invents computational techniques useful for exhibiting the consequences of those hypotheses:

... in August 1684 .. Halley .. decided to visit .. Newton in Cambridge ... After some pleasantries, he asked Newton what type of curve .. would be described by the planets orbiting under the inverse square law, and Newton immediately replied it would be an ellipse – and that he had proved it ... Halley asked to see the proof .. Newton said he couldn’t find it but would redo the demonstration and send it ...

This was impressive mathematics but only becomes science when we understand it as progress on astronomy and Kepler's laws

The role of philosophy in the psychology of the sciences cannot be dismissed: Ernst Mach's self-portrait "View from the left eye" (in which he looks out at the world past the frame of his own eyeglasses and past his own mustache) still retains its useful cautionary content a century later



Mach's views do seem to have helped motivate Einstein's development of relativity -- but it would be unwise to stop with that observation: Mach also opposed Boltzmann's atomism, on the grounds that atoms are unobservable, a view that would have been philosophically sound at the time but which in effect probably hindered the acceptance of statistical mechanics

The interchange between him and Stephen was fantastic. NDT is far beyond my comprehension, but I appreciate his knowledge and predictions. His laugh just has to be the best in the universe (or universes.)

Agreed!!! I really like the way he can explain (buzz word now is "un pack&quot something complicated into something I can understand.

"His laugh just has to be the best in the universe (or universes.)"

Yes!! If he looses his job he could do commercials.. Like the cola nut guy.... great voice..

If you get a chance get his latest book "Astrophysics for people in a hurry" . It's great.. makes you think... I like his "cosmic prospective" and it makes my transition between the infinities easier to understand...
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Embarrassed that I didn't know about it before Colbert referenced it.

NDT.

it is that the creativity is exercised in ways that lead to effective and usable calculations and to testable predictions

When you show me the calculational methods and the testable predictions, then I grant that the thesis is scientific (though not necessarily true); when you show me that the calculational results and the predictions match actual evidence, then I grant that there may be evidence for the thesis

But before that, I call the whole enterprise speculation, no matter how bold and exciting the ideas might seem

"Space is big. Really big. You just won't believe how vastly hugely mindbogglingly big it is. I mean you may think it's a long way down the road to the chemist, but that's just peanuts to space."

The set of all integers is infinite. So is the set of all even integers. But the set of even integers must be smaller than the set of all integers.

In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

https://en.wikipedia.org/wiki/Aleph_number

Decent article there, but you may have to do a little deep wading to get the totality of it.

At one time everyone thought Earth was the only planet until we learn out it was not.
At one time everyone thought the Sun was the only star in the galaxy until we learn it was not.
At one time everyone thought that the Milky Way was the only galaxy in the universe until we learn it was not.
Now why do we think that there is only 1 universe? For that matter why there will be 1 multiverse and etc and etc?

The universe is define as: all existing matter and space considered as a whole; the cosmos. The universe is believed to be at least 10 billion light years in diameter and contains a vast number of galaxies; it has been expanding since its creation in the Big Bang about 13 billion years ago.

However, this definition is only pertains to this cosmos / universe. Other universes could have different chemistry, physics, and uses other properties entirely. Atoms, matter, gravity, chemical makeup, and light could work radically different from what we are used too in these universes. If these other universes do exist, we would have to classify them different from our own.

<edited for grammar>