191–200 of 226 posts
I am a finitist and constructionist at heart. Sure, mathematical abstractions and infinite structures are fun to play around with.. But go ahead and actually provide me the list of all naturals. You can not. Ever.
But how did you come to this conclusion unless by assuming that there are infinitely many natural numbers?
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OK, but surely only because the exact value of 1 exists in the first place.
My first thought on reading your comment was to disagree and say no, we can have the exact value of 1, because we can choose our system of units and so we can make the square a unit square by fiat. A better way to dispute the unit square diagonal argument for the existence of sqrt(2) would be to argue that squares themselves are unphysical, since all measurements are imprecise and so we can't be sure that any two phy…
However, at each finite n we are still dealing with discrete quantities, i.e. integers and rationals. Even algebraic irrationals like sqrt(2) are ultimately a limit, and in my view the physicality of this limit doesn’t follow from the physicality of each individual element in the sequence. (Worse, quantum mechanics strongly suggests the sequence itself is unphysical below the Planck scale. But that’s not actually relevant - the physicality of sqrt(2) ultimately assumes a stronger view about reality than the physicality of 2 or 1/2.)
God created the rational numbers. The universe requires infinite divisibility, i.e. a dense set. It doesn't require infinite precision, i.e. a complete set. Our equations for the universe require a complete set, but that would be confusing the map with the territory. There is no physical evidence for uncountable infinities, those are purely in the imagination of man.
Earlier quoted context omitted.
My first thought on reading your comment was to disagree and say no, we can have the exact value of 1, because we can choose our system of units and so we can make the square a unit square by fiat. A better way to dispute the unit square diagonal argument for the existence of sqrt(2) would be to argue that squares themselves are unphysical, since all measurements are imprecise and so we can't be sure that any two phy…
You can say it’s exactly 1 plus or minus some small epsilon and use the completeness of the reals to argue that we can always build a finer ruler and push the epsilon down further. You have a sequence (meters, decimeters, centimeters, millimeters, etc) where a_n is the resolution of measurement and 5*a_(n+1) determines your uncertainty. However, at each finite n we are still dealing with discrete quantities, i.e. int…
> They were both put in a room and at the other end was a $100 and a free A on a test. The experimenter said that every 30 seconds they could travel half the distance between themselves and the prize. The mathematician stormed off, calling it pointless. The engineer was still in. The mathematician said “Don’t you see? You’ll never get close enough to actually reach her.” The engineer replied, “So? I’ll be close enough for all practical purposes.”
While you nod your head OR wag your finger, you continuously pass by that arbitrary epsilon you set around your self-disappointment regarding the ineffability of the limit; yet, the square root of two is both well defined and exists in the universe despite our limits to our ability to measure it.
Thankfully, it exists in nature anyhow -- just find a right angle!
One could simply define it as the ratio of the average distance between neighboring fluoride atoms and the average distance of fluoride to xenon in xenon tetrafluoride.
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The decimal places are just a way of representing it.
The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
Personally, I’d go with the sideline cut definition.
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The Planck length is at least around the right order of magnitude for things to get weird. If you have the position uncertainty of something be less that ~ a Planck length, and it’s expected momentum equal to zero, by Heisenberg position momentum uncertainty, the expectation of the square of the momentum is big enough that the (relativistic) kinetic energy is big enough that the Schwartzchild radius is also around th…
The right magnitude for things to get weird must be very small, but nobody can say whether that scale is a million times greater than the Planck length or a million times smaller than the Planck length. Therefore using the Planck length for any purpose is meaningless. For now, nobody can say anything about the value of a Schwartzschild radius in this range, because until now nobody succeeded to create a theory of gra…
So, like, I’m saying that if Einstein’s model of gravity is applicable at very tiny scales, and if the [p,x] relation continues to hold at those scales, then stuff gets weird (either by “measurement of any position to within that amount of precision results in black-hole-ish stuff”, OR “the models we have don’t correctly predict what would happen”)
Now, it might be that our current models stop being approximately accurate at scales much larger than the Planck scale (so, much before reaching it), but either they stop being accurate at or before (perhaps much before) that scale, or things get weird at around that scale.
Edit: the spins of fermions don’t make sense to attribute to something with extent spinning. The values of angular momentum that you get for an actual spinning thing, and what you get for the spin angular momentum for fermions, are offset by like, hbar/2.
All math is just a system of ideas, specifically rules that people made up and follow because it's useful. I'm so used to thinking this way that I don't understand what all the fuss is about, mathematical objects being "real". Ideas are real but they're not real in the way that rocks are. Whenever there's a mysterious pattern in nature, people have felt the need to assert that some immaterial "thing" makes it so. But…
The real question is whether 1 + 1 = 2 is true independent of us recognizing it. If the answer is no, then math really is just a system of ideas, and you’ve slipped into psychologism, where truth depends on minds. But take one thing and then another: you have two things. That’s true whether or not anyone notices. Some mathematics is a human system of ideas, but some of it isn’t. Arithmetic reflects real patterns in t…
You cannot justify this statement without equally justifying my position.
Say you conceive of a counterfactual world without any humans in it. You know that within this world there could be a rock and another rock, you understand that this would be two rocks, and so you are reassured that one and one is two, even though no one is watching within this counterfactual world.
All of this happened in your mind. All along, you were the observer of the supposedly unobserved world you conceived of.
You are the unavoidable human observer of any counterfactual world you conceive of. You intend the world to have no human observers, but your intention fails. It is impossible. The properties of a truly unobserved world are unknowable to you.
This is why the Enlightenment left Platonism behind centuries ago. We can't say what the world would be without us, because any attempt is not only constructed within the mind, but also contemplated and observed through the mind. You can't escape projecting your systems of ideas onto everything you think about.
Once this is taken into account, Platonism has no explanatory power and is nothing more than superfluous metaphysical mystification.
Earlier quoted context omitted.
The infinite number of decimal places is the definitional feature of a real number. No matter how's it represented they are still there and cannot be contained in our brains. We can say pi and hold the concept of pi in our heads, but not the actual number.
No, it really isn’t. The real numbers can be constructed in a number of ways, and it is more common to define them as either Dedekind cuts, or equivalence classes of Cauchy sequences of rational numbers. Personally, I’d go with the sideline cut definition.