joigus

Sorry, I meant "the derivative of x4 at x=5 is indeed 4*53" I hope that was clear... If it wasn't, please tell me. I'm sure you can.

No. It means you can't take x to be 5, or any other particular value. It must be a variable (varying, non-fixed) quantity. So the derivative of x5 is indeed 5x4, while 4*54 'is nothing of' 55. And the derivative of x5 at x=5 is indeed 4*54 But that is not what you said... It seems as if you're getting ahead of yourself. Maybe you need a good calculus book --like Spivak--, instead of calculus for dummies.

That must be it.

There is no evolutionary point in any change. Evolution comes into play when changes become stable through time. Today's slang is not the same as 18th-century slang.

Doubtful. That is plain wrong.

Looks like you're having trouble with the properties of powers, not with antiderivatives. Are you familiar with xn+m=xnxm? x-posted with @Genady

This is pseudoscience.

Very good questions indeed. The only proof of a scientific theory is experiment. Theory by itself doesn't allow us to prove a theory right, but it does allow us to prove it wrong. My advice would be to try to master trigonometry and calculus first. Also physics and chemistry, of course. Then algebra, geometry, topology... the works. Quantum mechanics, relativity --both special and general--, quantum field theory. Once you understand general relativity and quantum field theory, it's possible to understand why superstrings are perhaps worth considering. That's the pathway in a nutshell. You're allowed to enter a 'room' before you've completely understood the contents of the previous one. Otherwise it would take several lifetimes.

Sure it is. In fact, given enough context, a complete reference to the terms could become unnecessary, and saying 'I see' or 'I don't think so' could be enough to make clear what one means. If you think about it, we use elipsis most of the time when we are with family or close friends. They know what we mean. When I said syntax must have been present very early on, I meant that even in the first stages of development of language there must have been a very simple set of rules (subject)+(action)+(object) --or inverse order--, (subject)+(be)+(attribute). I don't think that crude pointing at things and naming could have been going on for much long. Linguists call this proto-syntax. It is, no doubt, speculation on the part of linguists --as language leaves no fossils--, but a very reasonable one. So obviously all words are more words couldn't be farther off the mark.

Or said index finger, but only when it's pointing to a rock. I think syntax is necessary very early on.

You would think so, but once a symbol has been introduced, it seems to acquire a 'combinatorial' life of its own, so to speak, that would make it very difficult for anyone to try to guess the meaning from just watching the symbol. In this respect, I found this talk by David Perlmutter very interesting: It seems to suggest that this direct association between meaning and symbol is but an initial cue, and complicating factors come into play later. Particularly interesting are his comments on how nearly indistinguishable the two symbols for 'Canada' and 'Jom Kippur' are in ISL.

De Saussure already observed that symbols are arbitrary --except probably for onomatopoeia--, while associations of symbols are not. OP seems to be confusing 'arbitrary' with 'silly', and not taking the second step --associations-- at all. Let's see what they have to say for themselves.

Very interesting. Here's a cute YT video on the 'mechanical' version of it, (https://en.wikipedia.org/wiki/Braess's_paradox#Springs) Sorry that it's a bit off-topic.

Very good example of,

I'd say that your feel of how slowly something grows largely depends on how often and how intently you observe it. Remember that saying, 'it's like watching grass grow'?

Interesting...

I think what @Genady means is space is not some "stuff" of which you can put in more of it. I agree with pretty much everything else he --and others-- have said here. You can't use naive addition of velocities to "cancel out" receding speed of galaxies with speed of light. Both have dimensions of length / time, but are very different things. One is an expansion parameter that applies to the whole of the universe as per cosmological standard model; the other is the speed of photons when they go "past your nose" so to speak. The latter is always c (a universal constant.) For photons reaching you from near the cosmic horizon, what you get is extreme Doppler shift, so they are lower and lower frequency (longer and longer wavelenght) the closer you get to this horizon. This horizon is of a kinematical nature, but it doesn't manifest itself as stopping the photons in their tracks, but as making them closer and closer to invisible. The other mistake that you're making --ignoring space expansion since those photons were emitted-- I also agree with. The fact that it's a mistake, that is.

Things obviously have changed a bit since 1977, but it's always nice to listen to Feynman afresh. This talk was unknown to me, and I still haven't got around to it, but Feynman never disappoints. Enjoy.

This calculation seems to be correct. Just in case anybody's interested, and because this thread has recently been revived, there's another definition that overlaps with that of a factorial (or gamma function). Namely, the falling and rising factorials. Although it's a generalisation in a different sense. The variable is not the "n" in n!

I thought this might be a meaty topic of discussion --based on the title--, instead I find what looks very much like a hall of fame of people with receding hairline through history.

I bet on lazy journalism. I recently read "prices of groceries going down" as synonymous of "inflation of groceries going down". Then corrected in 24 hours in follow-ups of same topic.

Agreed. Not a language issue to me. Just plain wrong. Dear @Z.10.46, there is a game in town that's very similar to what you're trying to play here. It's called asymptotics. From: https://en.wikipedia.org/wiki/Asymptotic_analysis In asymptotics we have to be very careful though. We never use the = sign. But we use an equivalence relation \( \approx \). If, eg, \( x \) is "very small", you're allowed to do things like \( \left( 1+x \right)^{2} \approx 1+2x \), but never things like \( x \approx 0 \), which lead to sorry mistakes. There is no such thing as "infinity" in asymptotics; only very big quantities, which must be either possitive or negative. Otherwise --like in your quite absurd, dimensionally inconsistent expression--, the "quantity" \( x \) defined by \( x + 2 = 1/(c-v) \) with c=v is: 1) +infinity? That is, the right limit, or, 2) -infinity? That is, the left limit. Of course infinity +2 = infinity, but also infinity + 3 = infinity, infinity +17=infinity-1/pi, etc. And -infinity+2 = -infinity, etc. "Infinity" is not a number. It breaks all the algebraic rules. Example infinity + x = infinity + y does not imply x=y. I like to think of infinity more like a topological entity (the boundary of the real numbers). I think that's the way in which modern mathematics we tend to look upon it. But that's another story. And Swansont's objection about units still stands.

My Dunning-Kruger effect sensor went off the scale here. Your handling of units and zero/infinity is appaling.

Also, you got this wrong --perhaps a typo. The famous regularisation of this infinite divergent sum by means of the eta function produces -1/12.