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Messages - ﮎingulaЯiτy

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Your post appears to be a solid breakdown and covers the basics quite well. However, if I was trying to explain the them to someone completely scientifically illiterate, I'd probably elaborate more on condition 2.

The notion of "explaining things better" may seem a tad arbitrary to a newcomer unfamiliar with scientific rigor. The strawberry hypothesis :) will be obviously inferior to the other theories because it is completely non sequitur. The mistake I see people making most often, is crediting a hypothesis due to its descriptive power (explanations made after the relevant observations). Any hypothesis with descriptive power would still be far inferior to models with predictive power (falsifiable explanations offered in advance of new empirical observations).

The strawberry example has neither descriptive nor predictive power so it may not prepare people for noticing the shortcomings of retroactive explanations, when they are looking for a "better explanation".

I'd also remark that theories are made even stronger by their applicable scope. Ex. If you've seen "A Beautiful Mind", John Nash won his Nobel Prize for the Nash Equilibrium, because his game theory approach worked in so many different disciplines besides economics. It created insights in topics he had never considered like sociology, politics, insurance, conflict analysis, fiances, etc. I recently watched a numberphile video in which squared squares' network graphs gained insights from electrical models of circuits with various amps.

Even under a smaller less-abstracted scope, we can also use consistency checks against other working theories tangential to your current hypothesis. It's like sculpting the shape of a puzzle piece and then testing how it fits with its neighbors in the puzzle. Ex. Successful biology theories suggest that the nano fireflies hypothesis would require an oxygen mixed atmosphere in the flashlight for them to breathe and to fuel their bioluminescent reactions. In contrast, our models of electrochemical reactions suggest an incandescent filament would rapidly burn out in the presence of oxygen.

Consequently, we can make a prediction about the construction of the flashlight prior to investigation. This means we can ascertain the accuracy of an explanatory model, by measuring its predictive success. If sealed part of the flashlight houses a vacuum (or at least the absence of an oxidizer; noble gasses work because they are inert), the incandescent filament hypothesis is credited. If oxygen is present, the nano-fireflies bioluminescence hypothesis would be credited.

...I suspect you essentially knew all of these things, but I took this thread as an invitation to try to collaboratively narrow down the definitions and criteria.

Ultimately this thread seems to dissect the tenets of rational investigation, but I'll try not to hijack your thread and ramble about why Occams Razor is valid, or how a phenomenon could be mysterious to some particular person, but there could be no phenomena mysterious of themselves. ("Mysterious answer" is a contradiction in terms, and worshiping a sacred mystery is just to worship your own ignorance.)

In disciplines other than mathematics, the term "theory" denotes the highest possible status an explanatory model can ever attain by rigorously proving itself.

Science & Alternative Science / Re: Infinity
« on: August 10, 2017, 02:09:42 AM »
Apologies for the bump. I updated the equivalent thread on the other FES site, and realized I should do the same thing here.

I have to recommend this 3Blue1Brown video which does a good job explaining analytic continuation (AC). It cuts through a lot of the mysticism that other explanations seem to encourage by glossing over details. AC essentially sidesteps our previous inability to deal with undefined infinities and more extrapolate the true behavior of functions regardless without caring whether or not they invoke infinity. Step by step, we can construct an intuitive visual explanation for the problem and how AC approaches it.

Personally, the visual explanation of AC seems analogous to a function where there is a hole in a straight line.

Consider the function: f(x) = 5x/x
That function's output is obviously a line, and it consistently returns f(x) = 5, with the exception of x = 0 where we get a "0/0" undefined error.

If we somehow didn't know how to pull common factors out of a numerator and denominator and change the function to be f(x) = 5, we could still look at the graph and see what the answer clearly would be 5 when x is 0, despite a calculator's inability to evaluate clashing infinities.

You cannot operate with highly divergent infinite series using the rules of convergent series: the results will be spurious.

Also, when using analytic continuation, you are actually substracting infinity from infinity to get something finite: again, an abstract mathematical operation with no connection to the real world.

I don't see how analytic continuation invokes conflicting infinities. The original problem does, but AC should dodge that problem, just like factoring my super simple example function doesn't actually divide 0 by 0. Let me know if you still perceive this as a problem after watching the video.

As for your concern that we could get spurious results, or lack real world applicability, please skip to 6:28 in fappenhosen's video:

Additionally, I recall posting a minutephysics video explaining how analytic continuation predicted the fine-structure constant (α), which has been continually confirmed over and over as the accuracy of experimentation improved enough to measure more significant digits. I can hunt that video down if it's really important to you.

There is only one solution to the problem, it is the solution nature agrees on, and arguably it is more coherent than saying the solution is "infinity". Infinities don't seem to appear in nature, but they are inherently built into how we model nature. I firmly believe infinity is a degeneracy of our mathematics models.

Food for thought.  :P

Science & Alternative Science / Re: Infinity
« on: June 25, 2014, 04:57:27 AM »
But Singy, infinity isn't even a real number. Why do you bother with things that don't exist? :-B

This. Numbers aren't real. A smart guy once told me so. Nothing made out of numbers actually exists.
Please send me a pm of your checking/banking account numbers and pins. Sounds like you wont miss them. <3

Science & Alternative Science / Re: Infinity
« on: June 25, 2014, 04:54:08 AM »
Are you asking us a question or just showing off?  :-B
Sharing something I found interesting, and leaving it open for questions/discussion. I expected for "no's" and "wtf's" like I did when I first came across it. I feel like I would have had to discover the proof to be a show off. Instead I'm reading other people's life work.

Science & Alternative Science / Infinity
« on: June 24, 2014, 07:55:35 AM »
I might as well post this on the forum where I still know some of the people.

Analytic continuation allows us to approximate an infinite series as a function. The more terms you add to the series, the more accurately you describe the function. So at the limit, when you have all the terms, that series can be considered equal to that function.

This allows us to assign meaningful numerical values to infinite series, even divergent ones.

For example, the infinite series "1 + 2 + 3 + 4 + 5... " can be evaluated. However, the result is disturbingly counter-intuitive. If you stop adding terms at any finite point, you'll have a larger and larger number as the result. However, if you "evaluate this after an infinite number of terms, you'll find the sum to actually be -1/12.

There are a variety of proofs for this, and this number is demonstrably a meaningful . This result is actually seen in physics. Furthermore this result is a foundation in string theory for the number of required dimensions.

Arts & Entertainment / Re: Now Playing (the Video Game Version)
« on: December 09, 2013, 07:50:42 AM »
It's basically a multiplayer thief version of Pacman.

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