Skeptics "reasoning" applied to mathematics

[Dragonfly]

~ Given that, and, that we're talking 'historical' trials re a 'fair' (ahem) die...ADD in 5 more 'dice'.

~ Would it really be expectable that the summed (across all 6) 'deviations' (of each one of the 6) would NOT cancel the individual deviations to the 'ideal' expectation? (Assuming 'long enough' doesn't mean that the machines broke down a few centuries from now.)

The deviations might well be systematic, due to the way the dice are manufactured. If there are (very small) deviations due to the distribution of the spots (for example the probability of 6 coming up is slightly higher than that of 1 coming up, or just the other way around), increasing the number of dice won't help. If on the other hand the deviations are random, with a larger number of dice they'll tend to cancel each other out, although you'll need in general more than 6 dice for that. But the canceling is still a probabilistic process, they don't cancel exactly.

~ Or, would you disagree that I'd have justification for 'certainty' that it would?

You'll never get mathematical certainty, but a probabilistic "certainty" can be good enough. A good example is the 2nd law of thermodynamics, which is in fact a statistical law. If you drop a glass on the floor and it shatters into thousands of pieces, you'll never see that those pieces the next moment jump back and form again a complete glass. There is nothing in Newton's laws that forbids them to do so, it's only that the probability that it will happen is so extremely small that even if you'd repeat the experiment continuously during billions of years the probability that you'd ever see it is still vanishingly small. Insurance companies can make a lot of money using statistical probabilities.

PS: This expectation of divergence-findings is itself a 'theoretical' idea of reality diverging from math expectations, isn't it?

It's also what we find empirically in countless similar observations. Mathematical descriptions are always idealizations, which may be accurate enough for all practical purposes, but they are still idealizations.

PPS: You didn't really answer my last question in my previous post.

Sorry. You asked:

~ My question is: Can I justifiably say that I 'know' this with...100% 'certainty'? If not, how does one, um, determine the probability of 'certainty' applicable here?

I suppose it'll be clear now that you can't know this with 100% certainty. But 99.999% certainty is also very useful. Statistical theory enables you to calculate those probabilities from the observed data. This doesn't give you exact probabilities (to get an infinite number of decimals, you'd need an infinite number of trials), but estimates. But for all practical purposes those estimates can be accurate enough. Perhaps you should read some textbook of statistics. There are probably many of such books for psychologists, economists etc. which are not too technically (mathematically) demanding and which can give you some feeling for these issues.

[John Dailey]

Dragon:

~ 1st off, I've got to apologize. Your response prompts from me a...long one; ergo, given my probs re 'long' respondings-back (something about my firewall and when I hit 'send'; I'm limited to 2 paragraphs, more-or-less, per/post), I have to make such multi-post. That covered...

~ Thank you for suggesting that I -> 'read some textbook of statistics. There are probably many...which are not too technically (mathematically) demanding [appreciate that consideration. JD] and can give you [i gather: 'me'] some feeling for these issues.' --- I appreciate the thought you've given to what you perceive as my lack of knowledge, but, to paraphrase the old saying, "Been there, Read them."

~ However, I don't see how that's a relevent-at-all 'answer' (even if I never read such) to my last question which you 'responded' to, but, didn't really quite hit the mark of doing what I'd call 'answering'.

2Bcont

LLAP

J:D

[John Dailey]

Dragon:

~ I've reviewed a 'basic' book on probabilities (hopefully you see no diff, other than semantic, with this from 'statistics' as you specifically suggested), and find that a 1st basic explanation is in terms of coin-flipping. They argue about the Heads/Tails occurrences explaining that 'EVENTUALLY' (your word), regardless the empirical 'runs' of 10 consecutive H in a set of 20 coin-flips, at some point (given enough flips) there will have been the same quantity of H as T having occurred in the past set of flips. Ie, CONTRARY TO your idealized argument about a diff 'twixt calculation and physical 'trials', there'd be NO DIVERGENCE...'over-all.'

~ Of course, such determination of which is empirically-determinable as 'true'/Acceptable (divergence vs no-divergence) may depend upon whose (yours or theirs) idea of 'eventually' (time to stop flipping) is to be applied. Not sure how statistical/probablistic arguments apply there, but, I think we have a prob on finding the 1st page to agree on for starting from.

LLAP

J:D

[John Dailey]

Dragon:

~ Oh, yes...we were talking about dice, not coins. Ntl, I think you see my point. Probability-wise, no diff, right?

~ I grant (as I already did...theoretically...when I brought up the 5 other dice) your argument about a possible 'divergence' inherently occurring 'eventually' via the, can we agree (?) theoretical considerations re your argument about PHYSICAL dice.

~ BUT, you must grant that your argument about the dif 'twixt 'idealized' calculations of probabilities AND the expectation of empirical physical occurrences re a die-fall IS...itself...an 'idealized' argument about the difference; no? I mean, it really has not been 'empirically' established...sfarsino.

LLAP

J:D

[John Dailey]

Dragon:

~ IN-WHICH-CASE...

Though appreciative of your response-attempt to such, I'm STILL awaiting your 'answer' to my last 2 questions in my post #123; especially the LAST question.

MTFBWY (need it, methinks, you may)

LLAP

J:D

PS: I find your earlier comment "Mathematical descriptions are always idealizations, which may be accurate enough for all [?] practical purposes, but they are still idealizations" fascinating! So, the dif 'twixt an 'idealization' and 'all practical purposes' is...?

Edited June 7, 2007 by John Dailey

[John Dailey]

Dragon:

~ It's been a week or so since my last (and un-responded to) post, and, upon reviewing, I find a need for a new comment, in this thread, on this subject.

~ My original questions were predicated upon the QUITE CLARIFIED assumption of using a 'fair' die.

~ Your responses NEVER specifically questioned this assumption.

~ Your ultimate point, ntl, innuends and implies that there can be no such thing as a 'fair' die.

~ If, regardless the present-day technology of being able to move atoms from locations A-to-B (hence, 'theoretically' creating an atomically-'fair'-die), you accept this point, then why did you not specify this obvious assumption-disagreement 'twixt us in your 1st response, rather than lead me on into a (apparently arbitrarily) dropped, by you, discussion?

LLAP

J:D

Edited June 24, 2007 by John Dailey

[BaalChatzaf]

Dragon:

~ It's been a week or so since my last (and un-responded to) post, and, upon reviewing, I find a need for a new comment, in this thread, on this subject.

~ My original questions were predicated upon the QUITE CLARIFIED assumption of using a 'fair' die.

~ Your responses NEVER specifically questioned this assumption.

~ Your ultimate point, ntl, innuends and implies that there can be no such thing as a 'fair' die.

~ If, regardless the present-day technology of being able to move atoms from locations A-to-B (hence, 'theoretically' creating an atomically-'fair'-die), you accept this point, then why did you not specify this obvious assumption-disagreement 'twixt us in your 1st response, rather than lead me on into a (apparently arbitrarily) dropped, by you, discussion?

LLAP

J:D

Have you taken the Heisenberg Indeterminacy Principle into account? Said principle is empirically supported, has never been falsified and is at the base of quantum physics which has a very good empirical track record (the only record that counts).

Philosophical purity (other than internal consistency) is worth nada. Empirical support is everything. A theory is precisely as good as its predictions.

Ba'al Chatzaf

[Dragonfly]

First about that "arbitrarily dropped discussion": contrary to some of the other participants of this forum I don't have always time to follow all the discussions closely every day, so it may happen that there are hundreds of new posts since the last time I looked, and I can't read them all closely. In other words: I'd quite forgotten about this thread.

~ I've reviewed a 'basic' book on probabilities (hopefully you see no diff, other than semantic, with this from 'statistics' as you specifically suggested), and find that a 1st basic explanation is in terms of coin-flipping. They argue about the Heads/Tails occurrences explaining that 'EVENTUALLY' (your word), regardless the empirical 'runs' of 10 consecutive H in a set of 20 coin-flips, at some point (given enough flips) there will have been the same quantity of H as T having occurred in the past set of flips. Ie, CONTRARY TO your idealized argument about a diff 'twixt calculation and physical 'trials', there'd be NO DIVERGENCE...'over-all.'

As you state it, this is incorrect. Even if the coin is really fair, the probability that the quantity of H equals the quantity of T becomes smaller with increasing number of trials. On the other hand the proportion T/H gets closer to 1 with increasing number of trials.

A second point is that this example of a fair coin can only be an idealization. That may be good enough in practice, as the deviations from the ideal proportion T/H = 1 may be small enough. But when you're talking (like in your first post) about probabilities like .1666 ad infinitum (or for the coin a probability of exactly .5) then these will never be found in real life situations, if only while they would necessitate an infinite number of trials and even if you could throw the coin infinitely many times, you would not get those decimals as the really fair coin or die does not exist. Even a coin with the exactly correct distribution of atoms wouldn't work, not only QM but also classical, Newtonian physics makes it impossible, as you can't isolate the coin from all the external influences, for example the continuously changing gravitational field.

~ BUT, you must grant that your argument about the dif 'twixt 'idealized' calculations of probabilities AND the expectation of empirical physical occurrences re a die-fall IS...itself...an 'idealized' argument about the difference; no? I mean, it really has not been 'empirically' established...sfarsino.

No. Our experimental evidence indicates that all such models are idealizations, and it is also in accordance with our knowledge about physics, which tells us that empirical results must deviate from the idealized model if there are enough trials (for a small number of trials they can of course accidentally give an exact result, for example 2 coin throws with result H and T). So empirical evidence and our current knowledge about physics both indicate that there is a difference between the idealized coin and a real coin. Therefore if you maintain that there is no difference between the ideal coin and the real coin, the burden of proof is on you. Only, you never can prove it, as you'd need an infinite number of throws.

Though appreciative of your response-attempt to such, I'm STILL awaiting your 'answer' to my last 2 questions in my post #123; especially the LAST question.

~ My question is: Can I justifiably say that I 'know' this with...100% 'certainty'? If not, how does one, um, determine the probability of 'certainty' applicable here?

No, you cannot justifiably say that you know that with 100% certainty, as I have explained in my previous posts. The probability of 'certainty' is a finite number of decimals that you can experimentally determine divided by an infinite number of decimals that you'd need to have certainty, in other words: zero.

~ My original questions were predicated upon the QUITE CLARIFIED assumption of using a 'fair' die.

~ Your responses NEVER specifically questioned this assumption.

Yes of course they did. In my first response I wrote:

There is a difference between the ideal, theoretical die and a real, physical die. In the theoretical case the probability of a particular side coming up is from symmetry considerations exactly 1/6. But a real die is never completely fair, there will always be differences, even if they may be quite small (even the different distributions of the dots will have some influence).

Seems crystal clear to me, so I really don't understand your complaint.

[John Dailey]

Dragon:

~ Sorry for the comment about you're not 'getting back' soon enough, as if you were supposed to hanging ten over the keyboard (Get it? 'Surfing' and all? Am I superficial, or what? [Don't answer that!]) for my response. I'd been chronic in varied postings for a while, and didn't realize such after a few days' break 'till you commented on that. Nevah be it said again by moi.

~ Re your pertinent commentings :hmm: (thanx for being so comprehensive), uh, because of my parenthetical...lemme get back to you on those :ermm: ...later. All are 'food-for-thought', but, unclear as to which thought is mere idealization or empiricially established :question:

~ Great! From talking about a 'fair' macro-object die and presumably 'normal', basic, arguments about Probability Theory, now we're into H's Uncertainty Principle and QM.

~ Yep: 'Later.'

LLAP

J:D

Edited June 28, 2007 by John Dailey

[tjohnson]

I came up with a parallel of the following skeptic argument and I think it brings to light in a greater contrast the lunacy of such "reasoning".

The argument is that since we cannot know of variables that may affect a situation we cannot be certain of the conclusion we have based on the variables we do know of.

The lunacy of this type of reasoning is exposed in greater contrast when you apply it to mathematics.

Korzybski says it this way; In mathematics all characteristics are included in our definitions and so deductions, if performed correctly, always work. In any other language all characteristics cannot be included in definitions and so deductions only work relatively, no matter how well they are made. So you can't apply that statement to mathematics. FYI, it's possible to make 'certain' statements if they are framed in a negative fashion. So I can say for sure that the earth's shape is NOT a sphere, for example.

[BaalChatzaf]

Korzybski says it this way; In mathematics all characteristics are included in our definitions and so deductions, if performed correctly, always work. In any other language all characteristics cannot be included in definitions and so deductions only work relatively, no matter how well they are made. So you can't apply that statement to mathematics. FYI, it's possible to make 'certain' statements if they are framed in a negative fashion. So I can say for sure that the earth's shape is NOT a sphere, for example.

If you shrunk the earth down to the size of a billiard ball it would be smoother than any commercially available billiard ball. It would, however, be a tad oblate due to its rotation.

Ba'al Chatzaf

[Brant Gaede]

Korzybski says it this way; In mathematics all characteristics are included in our definitions and so deductions, if performed correctly, always work. In any other language all characteristics cannot be included in definitions and so deductions only work relatively, no matter how well they are made. So you can't apply that statement to mathematics. FYI, it's possible to make 'certain' statements if they are framed in a negative fashion. So I can say for sure that the earth's shape is NOT a sphere, for example.

If you shrunk the earth down to the size of a billiard ball it would be smoother than any commercially available billiard ball. It would, however, be a tad oblate due to its rotation.

Ba'al Chatzaf

Please don't do this.

--Brant

[John Dailey]

~ Let's see if I can 'integrate' these quotes...with the addition of one of my own quotings from one ref'd here...

(Dragon:) I suppose it'll be clear by now that you can't know ['X'] with 100% certainty. But 99.999% certainty is also very useful. [ questions: 'empirically', or merely 'idealizationally'? 'Macro'-probability, or, 'QM'-probability determination?]...But for all practical purposes those estimates can be accurate enough. [question: 'enough' for...talking about a 'fair' macro-die?]

(Baal:) Empirical support is everything. A theory is precisely as good as its predictions.

--- Question: what has QM and H's Uncertainty Principle 'predicted' about a macro-object...like a die...which has been 'empirically' supported? Are we delving into Schrodinger's poor cat territory here, or what?

2Bcont

LLAP

J:D

Edited August 15, 2007 by John Dailey

[John Dailey]

This example of a fair coin can only be an idealization. That may be good enough in practice...

--- ENOUGH! 'Practice' (aka 'empirical' trials) is what I started this whole thing about, and I see everyone giving 'idealized' arguments about how H's U-P must apply to macro-objects.

~ You guys don't see that your arguments are themselves 'idealized' with no empirical verification yet existing; it's all just...surmised? Indeed, with the contrary shown in 'empirical' trials in all (ok, 'non-QM' probability studies, if you wish; as in: 'empirical')? Jee-e-e-z.

~ To argue from H's U-P, which applies only to 'certainty' regarding position AND momentum being simultaneously (sorry, Al) determinable to an arbitrary/optional measuring point, to macro-objects such as dice (or Schrodinger's Cat), is to carry an 'idealized' implication...too-o-o-o far; especially when one's going to add in arguments about 'empirical' studies.

LLAP

J:D

[John Dailey]

~ Oh, yeah: *my* final 'quoting'...

gen-seman brought up Korzybski as someone worth considering in this whole subject. I'm not that familiar with the noted writer/theorist, but, one thing I do remember is reading something he once said, and with which I wholeheartedly agree (and suspect that Dragon might be tempted to also, when in a 'macro' mind-set)...

"A difference which makes no difference, is no difference." (aka: one worth considering, debating, arguing, etc.) --- *my* emphases.

LLAP

J:D

[Dragonfly]

This example of a fair coin can only be an idealization. That may be good enough in practice...

--- ENOUGH! 'Practice' (aka 'empirical' trials) is what I started this whole thing about, and I see everyone giving 'idealized' arguments about how H's U-P must apply to macro-objects.

~ You guys don't see that your arguments are themselves 'idealized' with no empirical verification yet existing; it's all just...surmised?

Nonsense. The arguments are based on very solid empirical evidence.

Indeed, with the contrary shown in 'empirical' trials in all (ok, 'non-QM' probability studies, if you wish; as in: 'empirical')? Jee-e-e-z.

Where did you get that crazy notion?

~ To argue from H's U-P, which applies only to 'certainty' regarding position AND momentum being simultaneously (sorry, Al) determinable to an arbitrary/optional measuring point, to macro-objects such as dice (or Schrodinger's Cat), is to carry an 'idealized' implication...too-o-o-o far; especially when one's going to add in arguments about 'empirical' studies.

There is nothing idealized about Heisenberg's uncertainty relation, it has been verified countless times. With regard to the die: under normal circumstances the irregularities of the die and its environment will cause deviations of the "ideal" die that are much more important than the uncertainty relation. However, the point is that even if you were able to make an atomic perfect die and could throw it under idealized circumstances, the results wouldn't be that of the ideal die, while in that case Heisenberg still would put a spoke in the wheels.

[Guyau]

Following on

http://www.objectivistliving.com/forums/in...amp;#entry27765 (*),

I mentioned anisotropy measurements:

"Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Observations: Implications for Cosmology" by David Spergel et al. (Jan 2007).

This report of recent observations concerning isotropy in the cosmic background radiation is available at the NASA site, under the WMAP Overview, Three-Year Papers.

We have now:

“Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation”

E. Kamatusu et al.

http://lambda.gsfc.nasa.gov/product/map/dr...p_5yr_cosmo.pdf

Region of note, adding radio astronomy to microwave:

Is large cold spot a void? (no ordinary matter, no dark matter, only E-M radiation and dark energy)

http://webusers.astro.umn.edu/~larry/void/

http://webusers.astro.umn.edu/~larry/coldspot.pdf

Is large cold spot a texture? (a type of topological defect in spacetime, from Higgs field symmetry-breaking phase transition)

http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.5737v1.pdf

http://www.damtp.cam.ac.uk/cosmos/viz/movi...exturesciam.pdf

~~~~~~~~~~~~

*Background: http://www.objectivistliving.com/forums/in...amp;#entry27470

http://www.objectivistliving.com/forums/in...amp;#entry27475

Edited July 6, 2008 by Stephen Boydstun

[BaalChatzaf]

You admit that we can be certain that 1+1=2 yet you cannot admit that we can be certain that when two (lets keep it simple for discussions sake and say two) variables interact within reality we can be certain of their consequence. Rather or not there are unknown variables in either case misses the fundamental point. In one case you are allowing for contextual certainty and in another you are not.

Pure mathematics is entirely abstract. It is the mathematician who defines the objects by postulates and states the rules. Physical nature is not involved. In a purely abstract context there is NO interaction with physical reality. Where the mathematics is interpreted, i.e. mapped onto a physical structure or process there is no guarantee that the logical conclusions drawn from the mathematical basis, will coincide with physical reality.

In particular, there is no guarantee that a scientific theory that has been well supported by many experiments will not be falsified under some (as yet unknown) condition. Physics offers no guarantees of truth.

Ba'al Chatzaf

[BaalChatzaf]

. . .

That word, singularity, is one hell of a word, isn't it? Why does it sound like God to me? Maybe because it is a concept for something that exists but does not exist? And something that is nothing, but which created all the rest? That's a pretty shaky premise if that is the end-point of deducing things from observation wedded to the Theory of Relativity.

Michael

A singularity is a condition or region in which our mathematical laws don't apply or break down. There is nothing religious about a singularity. Consider the function 1/x. The function is not defined for x = 0, so x = 0 is a singularity of the function.

See http://en.wikipedia.org/wiki/Singulariti_(math)

Cosmological theories of the Origin of Everything are necessarily speculative since we cannot create a Cosmos de novo in our laboratories.

By the way, Paul Steinhardt, a physicist at Princeton University has a theory wherein the Big Bang is not singular and there is a "before". Steinhardt and Turok's theory avoids the problems with Hoyles "forever cosmos" and has good agreement with cosmological data including early and late acceleration.

For a non-mathematical overview of this theory see:

Endless Universe: Beyond the Big Bang (Hardcover) by Paul Steinhardt and Neil Turok.

Details on the book can be found at

http://www.amazon.com/Endless-Universe-Bey...7267&sr=8-1

Ba'al Chatzaf

Edited July 26, 2008 by BaalChatzaf

[Guyau]

Scientific Cosmology

http://www.objectivistliving.com/forums/in...amp;#entry27470

Update:

Martin Bojowald’s Scientific American (Oct 08) Article

http://www.sciam.com/article.cfm?id=big-bang-or-big-bounce

Abhay Ashtekar’s Assistance

http://cgpg.gravity.psu.edu/people/Ashtekar/articles.html

[Guyau]

.

Expansion Forever

.

[tjohnson]

.

Expansion Forever

.

From the above article;

Professor Priyamvada Natarajan of Yale University, a leading cosmologist and co-author of this study, said that the findings finally proved "exactly what the fate of the Universe will be".

We will never know "exactly what the fate of the Universe will be". To say things like this is to confuse science with omniscience.

[BaalChatzaf]

Does anyone have any kind of real proof that the universe started at some point?

Michael

If by "real proof" you mean certain proof, the sort of proof that is used in mathematics, then the answer is no. All we have are some well corroberated hypotheses that so far have made good predictions and have not yet been falsified. That is as close to "real proof" as science gets.

Ba'al Chatzaf

[BaalChatzaf]

And the theory of relativity is solid science.

Dragonfly,

So? So have been other theories in the past like Newtonian physics. That doesn't make them explain things like quantum physics.

Is the Theory of Relativity the proper standard for proving that the universe had a beginning?

Michael

No. The theory of general relativity without Einstein's fudge factor indicates that the cosmos must either be expanding or contracting. It cannot be stable without an ad hoc factor being inserted into the theory.

The empirical evidence that the cosmos is expanding came from Hubble's observations initially.

Once Einstein realized what Hubble had observed he characterized his addition of the stabalizing fudge factor as his "biggest blunder".

Ba'al Chatzaf

[Guyau]

.

Expansion Forever

.

From the above article;

Professor Priyamvada Natarajan of Yale University, a leading cosmologist and co-author of this study, said that the findings finally proved "exactly what the fate of the Universe will be".

We will never know "exactly what the fate of the Universe will be". To say things like this is to confuse science with omniscience.

Thomas,

There are quotations from Professor Natarajan in the following two reports that strongly suggest she was misquoted in the BBC piece.

esa

Yale

It does strike me that one day people will know the fate of the universe as surely as today we know the fate of our own sun. In both cases, there are the layers of evidence and tested theory for the conclusions, which are the context of knowledge of these conclusions, contexts carefully kept by science.