Irrational math

[HydrogenBond]

This idea came to me in another forum topic. I thought I would present my insight for expert consideration as a math thread. I will begin with a simple analogy. Say we had three people watching children playing in the playground. The first person is grouchy and sees the children as loud and annoying. The second person is having a happy day and sees the children at play as something positive and calming. The third person is paranoid and fearful and sees the child's play as an accident waiting to happen. In practical terms, all three are seeing the same data input, but each is selectively filtering the data through an emotional filter, thereby biasing the entire data set into their own subjective point of view.

 

Relative to statistics, each can run a statistical study to demonstrate the validity of their claim. The first can correlate how random noise is stressful and can lead to irritation. The next can run a statistical study to correlate how watching child play can have a soothing affect. The last can run or sight statistical studies that show the hazards of the playground. In each case, the irrational emotional filter can be supported with irrational math.

 

I am not saying that statistics always supports irrational states. For example, it can be used to optimize a product and minimize defects, which is based on the rational objective of maximizing productivity. But there appears to be a zone where statistics can be used to support states of mind that are not rational, since it is a form of irrational math. Does anyone know where this subjective transition zone begins and ends? Statistics is such an important tool of science, I would hate to think some of our important theory is irrational. Statistics is not just rational or irrational, but appears to be useful for either objective.

 

If we go back to the playground, with pure objectivity, without any irrational emotional filter, one might conclude that all three points of view better reflect the sum of the data. There are wild kids out of control making too much noise. There are nice kids who are having fun, who are fun to watch. There are also risk takers and spaced cadets, who are accidents waiting to happen. There is no need for a statical study to prove this, since it is objective. The irrational math only becomes useful if we wish to bias the data set with some level of human subjectivity.

 

I never really liked statistics because of this irrational zone, which nobody has ever attempted to investigate to determine the rational cut off. The usefulness of statistics in other areas of science has led to the irrational conclusion that if the statistical study is done by the book, that is all that it necessary to call it valid science, even if it is irrationally biased.

 

What it comes down to is a lack of rational precursor thinking to filter the premise to make sure it is not based on an irrational filter. One can have a theory and is not required to include other points of view, but can go right for the statistics, since it is flexible enough to support irrationality.

[Qfwfq]

What's your definition of irrational math?

[LaurieAG]

What's your definition of irrational math?

 

Hi Qfwfq,

 

I think HydrogenBond is talking about the symbolism of irrational constructs that appear when we analyse everyday situations from varying points of view.

 

When you introduce irrational constructs in maths you are using them to counter out an anti irrational construct (i.e. i x i = -1) within the higher level structure of the problem itself.

 

An Israeli academic has recently put forward an explanation for certain visions recorded in the old testament being as a result of the ingestion of hallucinogenic parts of the acacia plant.

 

This explains how situations akin to genocide can be reconciled by people with a certain viewpoint, a certain vested interest, even though it appears that they are ordered to do it by a higher authority (by voices in someones head no less). 'And if you do not kill them all they will be like a thorn in your side forever, and I will end up doing to you what I planned to do to them'.

 

I think that HydrogenBonds conjecture is a good application of the scientific process to psychology because it highlights how one irrational extreme does not cancel out its extreme opposite .

 

IMHO, there is a problem with the introduction of irrational structures in maths to counter inherent reverse irrational structures. i.e. you shouldn't have either, at the beginning, in the end, or anywhere in between.

[HydrogenBond]

If look at statistics, it follows laws of math, which by its very nature, leave a level of uncertainty with respect to cause and affect. We know the coin will fall 50% of the time on heads, over time. But we don't know when. Will it be after two tosses, sometimes. How about 50 tosses, sometimes, etc.. This world view is no longer rationally certain because it is ruled by chaos. Chaos by it very nature is irrational, since it can bring affects that we can not yet explain with reason. What if E=MC2, worked only 93% of the time, plus or minus 6%. Now we can do all kinds of things that are not even rational. But when E=MC2 things get thinned down, removing subjective inflation.

[LaurieAG]

If look at statistics, it follows laws of math, which by its very nature, leave a level of uncertainty with respect to cause and affect. We know the coin will fall 50% of the time on heads, over time. But we don't know when. Will it be after two tosses, sometimes. How about 50 tosses, sometimes, etc.. This world view is no longer rationally certain because it is ruled by chaos.

 

Hello HydrogenBond,

 

This 'world' you describe is governed by infinity, because only over an infinite number of tosses will the result be exactly 50 50.

 

This is a good example of where a structural deficiency, or irrational aspect of the design, causes all outcomes to appear chaotic even if they do appear logical.

 

So, if you structure a problem that requires infinity to gain an absolute result, don't be surprised if the only results that you can guarantee are uncertain ones.

 

Time goes on both ways forever

despite all mortal human endeavour

infinity will be reached, never ever.

 

Therefore if you structure a problem from a 'god' perspective (i.e. not a mortal perspective because we mortals will never experience infinity) don't be surprised if your answers are uncertain in the mortal perspective, because in these cases the usual mortal answer is 'god only knows'.

[Qfwfq]

We know the coin will fall 50% of the time on heads, over time. But we don't know when. Will it be after two tosses, sometimes. How about 50 tosses, sometimes, etc..

...

because only over an infinite number of tosses will the result be exactly 50 50.

The number of heads in a given number of tosses is an example of the binomial distribution. Although the average number of heads tends toward half the number of tosses, with this number increasing toward infinity, the standard deviation will be increasing proportionally to the square root of number of tosses and the probability of heads being exactly half of tosses tends toward zero.

 

Just a little remark....;)

[CraigD]

If look at statistics, it follows laws of math, which by its very nature, leave a level of uncertainty with respect to cause and affect. We know the coin will fall 50% of the time on heads, over time. But we don't know when. Will it be after two tosses, sometimes. How about 50 tosses, sometimes, etc. This world view is no longer rationally certain because it is ruled by chaos.

As Qfwfq notes, the count of heads or tails after any number of tosses is an example of the binomial distribution. Explicitly, the probability that we will have the same number of heads as tails after [math]2n[/math] tosses is

 

[math]\prod_{k=1}^n \frac{n+k}{4k}[/math]

(Note that the product symbol means [math]\prod_{a=1}^3 a = 1 \cdot 2 \cdot 3[/math])

So the probability of 50% heads after 2 tosses is [math]\frac{1+1}{4 \cdot 1} = \frac{1}{2}[/math],

 

after 20 tosses, [math]\frac{10 +1}{4 \cdot 1} \cdot \frac{10 +2}{4 \cdot 2} \ldots \frac{10 + 9}{4 \cdot 9} \cdot \frac{10 +10}{4 \cdot 10}= \frac{46189}{262144}[/math].

 

There is nothing chaotic (or even irrational, in the mathematical sense ;)) about this! If we know the initial conditions exactly – a fair coin and starting counts of heads and tails at zero – we can exactly predict the probability of having exactly the same number of heads and tails after any number of tosses. If our knowledge of initial conditions is inexact – for example, our coin is slightly biased, but we don’t know exactly how much – our predictions are slightly inaccurate. The greater our uncertainty of initial conditions, the greater the uncertainty of our predictions.

 

This relationship between certainty of initial conditions and accuracy of predicted outcomes is characteristic of a non-chaotic system. In contrast, a chaotic system is one in which even a very small uncertainty of initial conditions produces a large uncertainty of predicted outcomes.

Chaos by it very nature is irrational, since it can bring affects that we can not yet explain with reason.

When used in a philosophical or religious context, chaos can refer to a lack of reason and order, but when used in a mathematical one, it does not. I don’t think I can summarize the mathematical concept of chaos better than quoting from the wikipedia article “chaos”:

Mathematically,

chaos

means an aperiodic deterministic behavior which is very sensitive to its initial conditions, i.e., infinitesimal perturbations of initial conditions for a chaotic dynamical system lead to large variations of the orbit in the phase space.

 

In lay terms, chaotic systems are systems that look random but aren't. They are actually deterministic systems (predictable if you have enough information) governed by physical laws, that are very difficult to predict accurately (a commonly used example is weather forecasting).

The confusion of the religious/philosophical and mathematical usages of the term chaos is, I think, a common source of confusion, which I believe Hydrogenbond is suffering from as exhibited by his posts on the subject in this and other threads.

What if E=MC2, worked only 93% of the time, plus or minus 6%.

According to the best current theories for predicting physical phenomena, something like this is actually the case, but with “plus of minus”s many orders smaller that the examples 0.06.

 

The conventional interpretations of quantum physics include small uncertainties in the position and velocity of fundimental particles, and thus, of collections of these particles up to the scale of macroscopic bodies and systems to which equations such as [math]E = m c^2[/math] are commonly applied. However, like the previous coin tossing example, the aggregation of fundamental particles into the macroscopic is not a chaotic system, so the uncertainty on a “quantum level” is not significant on a macroscopic level.

 

This does not mean it is not present, only that the probabilities of bizarre events, such as my PC suddenly quantum tunneling across the room, are so small that all of the people who have or will ever live are exceedingly unlikely to ever witness such an event.

Now we can do all kinds of things that are not even rational. But when E=MC2 things get thinned down, removing subjective inflation.

I’m unclear as to what these “all kinds of things that are not even rational” and “subjective inflation” Hydrogenbond refers are, and suspect that the irrationality in most of the examples he could provide can be explained as a misunderstanding of their subject matter. For example, the statement

I never really liked statistics because of this irrational zone, which nobody has ever attempted to investigate to determine the rational cut off.

suggests to me a lack of appreciation for much of the discipline and literature of statistics, which seeks to more precisely define, predict, and understand uncertainty, and the huge amount of effort over many generations that has been made in this pursuit.

 

I fundamentally disagree with the assertion that mathematics and physics are unreasoning (we do well, I think, to avoid the term “irrational” in math discussions because of it prevalent use to describe numbers that cannot be defined as quotients of integers) and subjective. Although, in the practical course of using the tools of math and physics, it’s often necessary to use subjective hunches and intuitions where formal proof is beyond one’s immediate ability, a hallmark of good math and science is that it precisely states where such unproven assumptions are made.

[LaurieAG]

I fundamentally disagree with the assertion that mathematics and physics are unreasoning (we do well, I think, to avoid the term “irrational” in math discussions because of it prevalent use to describe numbers that cannot be defined as quotients of integers) and subjective. Although, in the practical course of using the tools of math and physics, it’s often necessary to use subjective hunches and intuitions where formal proof is beyond one’s immediate ability, a hallmark of good math and science is that it precisely states where such unproven assumptions are made.

 

Hi CraigD,

 

Good points and I agree that the maths definition of 'irrational' does not cover all aspects of what the general public would consider irrational like i the Imaginary unit - Wikipedia, the free encyclopedia.

 

It seems like some aspects of physics and maths provide relatively certain results compared with other aspects that provide relatively uncertain results.

[Qfwfq]

Explicitly, the probability that we will have the same number of heads as tails after [math]2n[/math] tosses is

 

[math]prod_{k=1}^nfrac{n+k}{4k}[/math]

Just in case anyone is in the mood for using the above expression, I've a mild suspicion ;) that the 4 in the denominator should be replaced by [imath]2^{2n}[/imath] as follows:

 

[math]\prod_{k=1}^n\frac{n+k}{2^{2n}k}[/math]

[LaurieAG]

Just in case anyone is in the mood for using the above expression, I've a mild suspicion ;) that the 4 in the denominator should be replaced by [imath]2^{2n}[/imath] as follows:

 

[math]prod_{k=1}^nfrac{n+k}{2^{2n}k}[/math]

 

Hi Qfwfq,

 

After looking through Wiki for all asociated links with imaginary numbers/units, infinity (unbounded improper integrals), electrical physics, Lorenzian manipulations, Poincare sections, Minkowski space, Schroedingers equations etc and virtually everything to do with maths of the different types of relativity (and a lot of other related stuff along the timeline) and there appears to be one common factor involved in them 'i'.

 

The main thing about i is that it has an interesting repeating power sequence (of 4, due to its incestuous relationship with itself) which gives even more interesting variations when differentiated/integrated in applied calculus.

 

While you have considerable experience with maths, have you noticed this common factor?

 

I wonder if 'quantum' uncertainty is just an artifact of the manipulation of i because it is the common factor (maybe not stated explicitly but it's there if you dig deep enough) in the paper trail?

[CraigD]

[math]\prod_{k=1}^n \frac{n+k}{4k}[/math]

Just in case anyone is in the mood for using the above expression, I've a mild suspicion ;) that the 4 in the denominator should be replaced by [imath]2^{2n}[/imath] as follows:

[math]\prod_{k=1}^n\frac{n+k}{2^{2n}k}[/math]

 

Though it’s a bit of topic, I can’t resist responding. :)

 

Try systematically counting for some small values of n

n=1: hh ht* th* tt, [math]\frac24 = \frac12[/math]

n=2: hhhh hhht hhth hhtt* hthh htht* htth* httt thhh thht* thth* thtt tthh* ttht ttth tttt, [math]\frac6{16} = \frac38[/math]

etc.

This agrees with

[math]\prod_{k=1}^1 \frac{n+k}{4k} = \frac{1+1}{4\cdot1} = \frac24 = \frac12[/math]

and

[math]\prod_{k=1}^2 \frac{2+k}{4k} = \frac{2+1}{4\cdot1} \cdot \frac{2+2}{4\cdot2} = \frac{12}{32} = \frac38[/math]

and with

[math]\prod_{k=1}^1\frac{1+1}{2^{2 \cdot 1}k} = \frac24 = \frac12 [/math]

but not with

[math]\prod_{k=1}^2\frac{2+k}{2^{2\cdot 2}k} = \frac{2+1}{16\cdot1} \cdot \frac{2+2}{16\cdot2} = \frac{12}{512} = \frac{12}{512} = \frac3{128} [/math]

 

Note that the formula gives the probability of getting the same number of heads and tails after exactly 2n fair coin tosses, not the probability of getting the same number of heads and tails after 2n tosses when you stop tossing upon getting the same number of heads and tails. The formula for this probability appears to be (rather amazingly, to me):

 

[math]\frac{(2n)!}{(2n-1) 4^n(n!)^2}[/math]

[Qfwfq]

Darn I mistranslated from the usual factorial form to your friggin' PI form! :)

 

...not the probability of getting the same number of heads and tails after 2n tosses when you stop tossing upon getting the same number of heads and tails.

I suppose you mean excluding cases where the subsequence of the first 2m tosses evens up, with m < n. I don't understand the 2n - 1 factor either, I'd have to figure it out.

[LaurieAG]

What it comes down to is a lack of rational precursor thinking to filter the premise to make sure it is not based on an irrational filter. One can have a theory and is not required to include other points of view, but can go right for the statistics, since it is flexible enough to support irrationality.

 

Hi HydrogenBond,

 

An 'imaginary' filter wouldn't be needed for quantum stuff when you consider the outputs from the 'imaginary' practical examples given in electrical physics. i.e. if the electrical physics answer is 120 Volts 90 degrees out of phase, surely other non applied maths based on imaginary numbers must have an 'out of phase' type equivalent as part of their answers.

 

It's irrational that nobody has explained their theoretical answers in these terms.