Butterflys and Gods

Tao_Equus

Interfaith Forums
Messages
5,826
Reaction score
4
Points
0
Location
Edinburgh, scotland
This equation dz/dt=xy-z will be known to most of you as the Butterfly Effect. I will here attempt to demonstrate that the implication of this equation on the nature of reality leaves no possibility for the existence of a creator bearing any resemblance to the concept of god, as preached, in the universe we observe.

Unlike the old Newtonian laws suggest at first glance there is fundamental unpredictability in everything. The butterfly effect, chaos, and unmeasurable variables exist in everything. The developing mathematical models that try to describe instability in anything can do so only in the most limited respect. This is because everything is influenced by so many variables that are in turn subject to their own network of variability which in turn are...ad infinitum.

Newtons laws of motion, for example, can predict the orbit of two bodies around each other from now until infinity. And we have used these physics to great advantage, they work. But the reason they work is because inherrant instability does not usually have time to manifest, but sometimes it does. Introduce a third orbiting body and Newtons laws no longer work. Sooner or later some unobservable variable will begin to influence the system and very rapidly, almost instantaneously, chaos will be introduced. For example I think the general perception is that astronomers can predict the orbits of the planets indefinitely by use of Newtons laws. This is not true at all. The complex interaction of all the bodies in the solar system allow for only relatively short predictions of position. We can only really make approximations. And they have margins of error attached to them.

We have easily adopted Newtons thinking as it fits well with our cultural history of a monotheistic creator god that designed everything in a mechanistic, almost clockwork, perfection. Thinkers like Laplace compounded our ignorance by promoting the idea that everything was measurable, understandable, reductable and ultimately could thus be controlled. I think many people I have debated with here think I am of that school, and I appreciate that I can give that impression, but look closely and it is far from true that I accept anything on its reducibility alone.

We have briefly discussed Chaos Theory here. And all who partook marvelled at the computer generated imagery that gave humanity, for the first time, the ability to visually comprehend infinity. If you recall from that thread somebody dug up the term "Gods Fingerprint" to describe the laws of chaos. This has had me thinking a lot. Wondering if the discovery of this law was indeed evidence for an active creator. In fact for me it became the most compelling bit of creator suggestive mathematics I had ever encountered. For at face value such an irrefutable pattern inbuilt into the form of everything seems like a code for creation. But it was my misunderstanding of what it is, and its influence on the material universe, that was flawed. In actuality it is powerful evidence for the non-existence of a creator.

Edward Lorenz, the scientist who coined the term "The Butterfly Effect" sadly died this year. His work on divergence shows the bigger truth of chaos theory. It is easy to get drawn into the beautiful symmetry of a Mandelbrot set and think there is order there. But such a set is artificial. It is the generation of infinite repetition of a closed system. But there is no such thing as a closed system.Nothing in the observable universe is alone and unaffected by a wide range of dynamic influences each with their inbuilt and unmeasurable divergent properties. A Mandelbrot set is a human construct of perfect chaos. But such perfection exists nowhere.

An all knowing creator god has an impossible task. It would have to know with exact precision the divergent potential and predict what every particle, and sub-particle, in the universe would not only do, but what its divergent influence on every other particle would be from the beginning till the end of time. You can imagine this is mathematical task of such scale that it has to be ruled out as a rational possibility.

To give an example of the chaos involved. Just to predict the turbulence within a cup of coffee when milk is added to it you would need to measure 10 to the power of 40 degrees of freedom. There is no computer that can yet make such accurate measurement. And that is just the internal dynamics. The thermal turbulence of the cup on its local environment takes it into numbers that are incalculable.

Now if you scale this up to the turbulence within a star, or a galaxy, or a galactic cluster, then you begin to see the scale of the mathematical calculations any creator would be faced with. It is tempting to argue that our universe is stable, that there must be laws we don't understand that allow a creator to short cut all these calculations. But that is not so. The more we look the more we realise that the universe we observe is wholly at the mercy of divergence. We date the universe at about 13.8 billion years old. I think it is pretty safe to say that this was the last time that chaotic divergence caused a massive breakdown in the stability of the laws of physics that existed prior to that time. Cosmologists are continually finding anomalous objects that defy their best predictions and again the reason for this is divergence allows no standard prediction to be made. They can generalise but there will always be anomalies that crop up due to inherrant instability.

What all this says is that we can infer laws from studying the universe. But no matter what the law is everything that is supposedly governed by it can have a divergent potential that breaks the law. This potential is unmeasurable. So there can be no creator. A creator would not dare create a butterfly, lest it destroy all its work.


tao
 
Hi Tao —

The one thing I have learned about science is, what is impossible this year often becomes possible next.

In short, science is constantly revising its findings, redefining itself in the face of new data. What was the case yesterday, is not the case today, and what is the case today, might not be the case tomorrow.

We are a generation caught in the termination of a scientific proposition ... I distinctly remember being told, 30 years ago, that today I would live a life of leisure, technology having taken the burden of day to dasy existence off my shoulders.

Wrong. Totally wrong. More technology than we ever imagined, less free time, more time-driven, less uncertainty, not more.

If I want to know a quantitative measure, the melting point of beeswax, the time it takes to fly at .8 the speed of light to a star ... then science is pretty useful.

On qualitative matters, I believe science has shown again and again to be next to no use whatsoever.

Therefore to say that something like an Absolute Deity cannot be, I regard as a classic case of science overstepping its boundaries.

The equation "because I cannot conceive it: it cannot be" is false, as much as science tries to insist it is not.

You can't explain it, and it certainly seems impossible, but then again, so do many things which, today, we simply take for granted.

Next year someone might uncover the 'universal theory of everything' in an equation so simple, that the Divine task will be shown to be easy.

And surely the most telling counter argument is that many, many scientists are holders of a religious belief?

Whilst physics, for example, continues to discover itself, requiring a complete restructure of itself, theology in the Catholic or Orthodox traditions, has been able to 'move with the times' for some 2,000 years without having to revise or rewrite its original statements ... they still stand.

So statistically therefore, theology is a safer 'science' in which to place one's faith, than physics.

Thomas
 
Thank you for your thoughts Thomas :)

I appreciate and agree with your observations on the fickleness of science when viewed over decades. But I would say if we did not see this evolution within science then it would be in an unhealthy state and would indicate that any progress in our understanding had come to a halt.

You can argue that religion, and in your case I have to presume you mean the Catholic religion, has and can adapt to the discoveries that science makes. But really this is as slippery a slope on which to position themselves as standing firm on ancient dogma. The science will change, and the effort to appear "up to speed" will mean the Catholic church will have to keep altering its position. This, of course, is a statement that the church does not really have a clue and is only trying to remain scientifically credible.

As for the physics of chaos, on which my OP is inspired, they are a relatively new branch of mathematics that were discovered by accident by scientists that were not looking for them but believed in Laplacian reductibility. They were discovered amidst much denial and disbelief amongst the scientific community but were elevated by their incontrovertible observational reality. This branch of maths is still new and it may be that we can discover more rules hidden in all that chaos, and we probably will, but the underlying principle of uncertainty, after a now huge quantity of observational data, is so fundamental that it is inconceivable that it is not a universal principle.

I do not have time to relate it to and list its implications in the other fundamental rules of relativity and quantum mathematics, but they are profound and far reaching. To my knowledge no paper has yet been published that has made an effort to marry this trinity together into a coherent whole. No doubt it will come though and I look forward to both it and the debate it will stimulate.

Science cannot disprove god. But it can produce evidence that makes the existance of God ever more unlikely, and does so. I refuse to get drawn again into another debate about how many scientists are believers. Its an irrelevancy. And you know fine well that they number only a small percentage.

The study of uncertainty and divergence is creeping into just about every branch of science. It is as yet a baby. But its implications are startling, if uncomfortable. Even this economic crisis we are in was predicted some time ago by people studying divergence in number systems. It is the murphys law of science. And from my armchair exploration of its rules I can and do see a universe in which there is either no creator or a kind of young schoolboy one that just throws everything in a test tube, gives it a good shake, and looks on to see what happens. That is no all seeing, all knowing god.

tao
 
Lol, as it stands a re-read reveals there is a glaring error in my OP regarding self-similarity or Mandelbrot sets. Do not have time to explain it just now but the error does not change my overall statement.

tao
 
Thank you for your thoughts Thomas :)

I appreciate and agree with your observations on the fickleness of science when viewed over decades. But I would say if we did not see this evolution within science then it would be in an unhealthy state and would indicate that any progress in our understanding had come to a halt.

You can argue that religion, and in your case I have to presume you mean the Catholic religion, has and can adapt to the discoveries that science makes. But really this is as slippery a slope on which to position themselves as standing firm on ancient dogma. The science will change, and the effort to appear "up to speed" will mean the Catholic church will have to keep altering its position. This, of course, is a statement that the church does not really have a clue and is only trying to remain scientifically credible.

As for the physics of chaos, on which my OP is inspired, they are a relatively new branch of mathematics that were discovered by accident by scientists that were not looking for them but believed in Laplacian reductibility. They were discovered amidst much denial and disbelief amongst the scientific community but were elevated by their incontrovertible observational reality. This branch of maths is still new and it may be that we can discover more rules hidden in all that chaos, and we probably will, but the underlying principle of uncertainty, after a now huge quantity of observational data, is so fundamental that it is inconceivable that it is not a universal principle.

I do not have time to relate it to and list its implications in the other fundamental rules of relativity and quantum mathematics, but they are profound and far reaching. To my knowledge no paper has yet been published that has made an effort to marry this trinity together into a coherent whole. No doubt it will come though and I look forward to both it and the debate it will stimulate.

Science cannot disprove god. But it can produce evidence that makes the existance of God ever more unlikely, and does so. I refuse to get drawn again into another debate about how many scientists are believers. Its an irrelevancy. And you know fine well that they number only a small percentage.

The study of uncertainty and divergence is creeping into just about every branch of science. It is as yet a baby. But its implications are startling, if uncomfortable. Even this economic crisis we are in was predicted some time ago by people studying divergence in number systems. It is the murphys law of science. And from my armchair exploration of its rules I can and do see a universe in which there is either no creator or a kind of young schoolboy one that just throws everything in a test tube, gives it a good shake, and looks on to see what happens. That is no all seeing, all knowing god.

tao

Have you read Drunkard's Walk? An opinion?

dmsingh
 
Have you read Drunkard's Walk? An opinion?

dmsingh

Not read a book of that name no but I am aware of and have looked at several stochastic process studies in the course of my delving into this subject. From what I do know a drunkards walk is described as X to the power of 3. What I am talking about is X thpo infinity. Even in a 3 body system divergence is enormous and fundamental instability soon appears. On a universal scale... well.... like I said...in a cup of coffee it becomes 10 thpo 40 just trying to measure the turbulence of pouring in some milk.

tao
 
It's a fallacy that new scientific theories sweep away everything that has gone before. For example relativity which is most like this in refuting the concept of absolute time still includes something close to Newtonian gravity as a limiting case. In this sense science is cumulative; to successfully advance a new theory you must explain everything that went before plus make new predictions.

Also Tao you must include all three equations in the Lorenz attractor (Lorenz attractor - Wikipedia, the free encyclopedia) - they are a coupled set - which I don't think you did in your opening post.

peace and blessings,


qj
 
Hi Tao —

I appreciate and agree with your observations on the fickleness of science when viewed over decades. But I would say if we did not see this evolution within science then it would be in an unhealthy state and would indicate that any progress in our understanding had come to a halt.
Oh, agreed ... But then, I think the same way about theology.

You can argue that religion, and in your case I have to presume you mean the Catholic religion, has and can adapt to the discoveries that science makes.
Easily. Christianity is not a 'science', so there should be no ground for conflict. We have come a cropper when trying to determine what science should or should not think, but I think it's fair to say that we can say that science is 'out of its depth' when it addresses the question of God.

But really this is as slippery a slope on which to position themselves as standing firm on ancient dogma.
I don't see why. I think one of the startling 'facts' of theology is no scientific discovery has required a revision or alteration of ancient dogma. Certainly we have reviewed and readdressed, but then that is precisely what science does.

The science will change, and the effort to appear "up to speed" will mean the Catholic church will have to keep altering its position.
Not so. The Catholic Church has never altered its position on matters of faith and morals. We have erred in matters of science ... but then, so has science ...

This branch of maths is still new and it may be that we can discover more rules hidden in all that chaos, and we probably will,
I happen to think so too.

but the underlying principle of uncertainty, after a now huge quantity of observational data, is so fundamental that it is inconceivable that it is not a universal principle.
But if there are rules, 'uncertainty' as we understand it might not be the case. Nothing is certain, for sure, but nevertheless we all hold that the laws of nature can be trusted ... as a scientist pointed out, if uncertainty was as real as the term implies, we'd never be able to get out of bed in the morning.

I do not have time to relate it to and list its implications in the other fundamental rules of relativity and quantum mathematics, but they are profound and far reaching.
I am sure ... within the context of their own domain.

Science cannot disprove god.
Nor can it prove God.

But it can produce evidence that makes the existance of God ever more unlikely, and does so.
No it doesn't ... you're assuming what God is like, even after admitting science can make no definite statement about the matter.

It can produce evidence to suggest that God is more than we can imagine or conceive, which is nothing different to what Monotheism has stated from the very beginning ... the more I understand science, the more I see God as something so much more than my humble notions.

I refuse to get drawn again into another debate about how many scientists are believers. Its an irrelevancy.
Tao! Nonsense! You state that a belief is incompatible with science ... yet scientists are believers ... not irrelevant at all, but it does undermine your argument.

And you know fine well that they number only a small percentage.
Tosh! Actually I haven't a clue, but working from those I know personally, there would seem to be a significant number. I have two 'constant' lecturers on my course, and one of those holds a Doctorate in Mechanical Engineering.

The study of uncertainty and divergence is creeping into just about every branch of science. It is as yet a baby.
Well, you'll catch up. We've been studying that, theologically, for about 1,700 years.

But its implications are startling, if uncomfortable.
Theologically, they are too. There are areas of investigation 'that dare not speak their names' as it were ...

Even this economic crisis we are in was predicted some time ago by people studying divergence in number systems.
Good. Of course, we predicted it a long time ago ... I can dig out the encyclicals that speak about just such the consequences we are now enduring.

In fact, Pope John Paul II and Pope Bendict XVI are well within their rights to say "We warned you, and you would not listen."

It's all part of the erroneous mindset we call 'modernism', which is not, as many think, a knee-jerk reaction to anything 'new', but a critique of assumed philosophical principles.

One aspect, for example, is that the idea of an ongoing linear development — 'progress' — is a fallacy. There is a difference between 'growth' and 'progress' ... Organic developments 'grow' and are cyclic, but because the world ignores this (universal principle and law), man-made structures work on a boom-boom-boom principle, and need periodic corrective restructuring, or they collapse in boom-and-bust.

The root of the problem is greed.

Thomas
 
Interesting conversation!

Wrong. Totally wrong. More technology than we ever imagined, less free time, more time-driven, less uncertainty, not more.

I don't think I agree with the underlined. I think there is truth in the adage, 'the more we know, the more we realize how much we don't know.'


The equation "because I cannot conceive it: it cannot be" is false, as much as science tries to insist it is not.

'Science' does not try to insist such a thing, but some people misapplying science *cough Dawkins cough* may try.

Next year someone might uncover the 'universal theory of everything' in an equation so simple, that the Divine task will be shown to be easy.
Ha! Good point. :D

And surely the most telling counter argument is that many, many scientists are holders of a religious belief?
I'm reading the excellent book Finding Darwin's God by one such scientist, Kenneth Miller.
 
I think it is an interesting paradox that we both require and can't abide in a deterministic world.

I think it is interesting that the observation of an event changes that event. It is interesting that aspects of physcis must take into account the uncertainty of a particle's simultaneous postion and momentum.

It is interesting that we build the pyramids of ethics on a concept of good that can only be supported by an unprovable absolute. Ethics (even humanistic ethics) require faith.
 
I don't think I agree with the underlined. I think there is truth in the adage, 'the more we know, the more we realize how much we don't know.'
No, nor do I! Rats! I got it the wrong way round! My error.

'Science' does not try to insist such a thing, but some people misapplying science *cough Dawkins cough* may try.
Quite right ... I was being unfair.

I heard this in a homily once ... and I can't begin to express how profoundly it struck me ... one of those 'oh, wow! moments of realisation ... I'm sure it's grown in my mind, but the priest was saying:
"... think of time, stretching back, all those lives, all those stories, all those events ... and time beyond that, into the far regions of the very birth of the cosmos ... now think of time stretching away ... infinitely away ... into the future ...

... think of space ... think of how far, and how long, we might travel, and never get there ... think of the sun, the stars, the nebulae, the galaxies ... think of all the wonders from the distant reaches of space and time our technologies have brought us ... think of the microscope, and the micro world, the nanoworld, the cosmos of space within the structure of the atom ...

... think of all that ... and then, think of the human mind, that can reach out to all that, and encompass it all within itself ... than can see, and comprehend, and wonder ... of all the wonders in creation, there is nothing anywhere near so wonderful as the mind of man, so small, and yet it can contain the impossible vastness of space and time ... of being"
Quite a homily, as I recall.

Then again, that another can see what you see, that you can share that vision ... what are the chances of that?

Thomas
 
Reminds me of this Eucharistic Prayer:

Lift up your hearts.
We lift them to the Lord.

Let us give thanks to the Lord our God.
It is right to give him thanks and praise.

Then, facing the Holy Table, the Celebrant proceeds

God of all power, Ruler of the Universe, you are worthy of
glory and praise.
Glory to you for ever and ever.

At your command all things came to be: the vast expanse of
interstellar space, galaxies, suns, the planets in their courses,
and this fragile earth, our island home.
By your will they were created and have their being.

From the primal elements you brought forth the human race,
and blessed us with memory, reason, and skill. You made us
the rulers of creation. But we turned against you, and betrayed
your trust; and we turned against one another.
Have mercy, Lord, for we are sinners in your sight.


Again and again, you called us to return. Through prophets
and sages you revealed your righteous Law. And in the
fullness of time you sent your only Son, born of a woman, to
fulfill your Law, to open for us the way of freedom and peace.
By his blood, he reconciled us.
By his wounds, we are healed.


And therefore we praise you, joining with the heavenly
chorus, with prophets, apostles, and martyrs, and with all
those in every generation who have looked to you in hope, to
proclaim with them your glory, in their unending hymn:



Celebrant and People

Holy, Holy, Holy Lord, God of power and might,
heaven and earth are full of your glory.
Hosanna in the highest.
Blessed is he who comes in the name of the Lord.
Hosanna in the highest.
 
It's a fallacy that new scientific theories sweep away everything that has gone before. For example relativity which is most like this in refuting the concept of absolute time still includes something close to Newtonian gravity as a limiting case. In this sense science is cumulative; to successfully advance a new theory you must explain everything that went before plus make new predictions.

Also Tao you must include all three equations in the Lorenz attractor (Lorenz attractor - Wikipedia, the free encyclopedia) - they are a coupled set - which I don't think you did in your opening post.

peace and blessings,


qj

Thank you for your observation QJ,

Of course you are correct. But there are so many fields of the scientific endeavour that I could also include, and are inter-related. Indeed there are few, very few, areas of thought that cannot be shown to rubbish the notion of a creator. But I am not a book writer, not even a scientist, and I have to limit myself to make a point concisely.

The leaps and bounds of science have been amazing to watch even in my lifetime. However we would do well not to be dazzled by the brilliance of so many men and women breaking so much new ground. But the question of fundamental instability is one we can measure in absolutely everything. In a universe with apparently so few universal laws this instability is one of the most profound and ubiquitous features we can observe. Anyone who has studied science with a modicum of depth will realise that the probability of a planet with just the right minerals/elements, with a moon just so positioned to drive tectonic convection to recycle them, going round a peaceful star in a stable solar system in just the right position on a galactic arm of a stable galaxy is pretty damned lucky place to be. But they also know this will not last long. The chances are that long before Andromeda smashes into the milky way, or the sun expands and evaporates the oceans some other calamitous celestial event will wipe us out. Our lifetimes, even our histories are not even eyeblinks in cosmic time. They are important to us of course, indeed they are everything to us, but are meaningless to the violent and hostile universe we inhabit. A universe full of writhing, twisting utterly incalculable unpredictability. A universe with no plan and certainly no planner.


tao
 
It is interesting that we build the pyramids of ethics on a concept of good that can only be supported by an unprovable absolute. Ethics (even humanistic ethics) require faith.

I refute with a passion that ethics has anything to do with religion. Ethics are no more than a formalised system of morality. Morality is a human trait, and can be observed in all social animals. It has nothing to do with faith or religion, no matter how much they would like to hijack it and call it their own.

tao
 
I heard this in a homily once ... and I can't begin to express how profoundly it struck me ... one of those 'oh, wow! moments of realisation ... I'm sure it's grown in my mind, but the priest was saying:
"... think of time, stretching back, all those lives, all those stories, all those events ... and time beyond that, into the far regions of the very birth of the cosmos ... now think of time stretching away ... infinitely away ... into the future ...

... think of space ... think of how far, and how long, we might travel, and never get there ... think of the sun, the stars, the nebulae, the galaxies ... think of all the wonders from the distant reaches of space and time our technologies have brought us ... think of the microscope, and the micro world, the nanoworld, the cosmos of space within the structure of the atom ...

... think of all that ... and then, think of the human mind, that can reach out to all that, and encompass it all within itself ... than can see, and comprehend, and wonder ... of all the wonders in creation, there is nothing anywhere near so wonderful as the mind of man, so small, and yet it can contain the impossible vastness of space and time ... of being"
Quite a homily, as I recall.

Then again, that another can see what you see, that you can share that vision ... what are the chances of that?

Thomas

This eloquent homily is typical of the irrational self deception of the possessor of faith. If you break it down it is apparent that it includes false reasoning and conclusion to arrive at a self-self-congratulatory boost to the ego. When religions are not busy telling you how evil a sinner you are they do this. The good cop/ bad cop of of the psyche out. We have no idea what the universe is. All we can encompass is our own ideas. We can only view a tiny fraction of what is. But it is important to the religions to paint us as special and gifted, made in gods image. But it is not true. We evolved from bacteria and viruses. We are apes with big brains that allow us the arrogance of our thoughts but they show us almost nothing of reality. So the preacher used pretty words, but they are meaningless at best. In truth they are not meaningless though, they are a poetic effort at deception to sway people to believe irrational falsehoods for the purpose of getting donations on the collection plate.

tao
 




@ Tao


From what I gather, your entire argument is based on the following:

An all knowing creator god has an impossible task. It would have to know with exact precision the divergent potential and predict what every particle, and sub-particle, in the universe would not only do, but what its divergent influence on every other particle would be from the beginning till the end of time. You can imagine this is mathematical task of such scale that it has to be ruled out as a rational possibility.
You think God, the Creator of the universe would be intimidated.... by math? ... By the same principles that He used to create the universe?

But no matter what the law is everything that is supposedly governed by it can have a divergent potential that breaks the law. This potential is unmeasurable. So there can be no creator. A creator would not dare create a butterfly, lest it destroy all its work.
No, it is unmeasurable for us...
 
Tao said:
An all knowing creator god has an impossible task. It would have to know with exact precision the divergent potential and predict what every particle, and sub-particle, in the universe would not only do, but what its divergent influence on every other particle would be from the beginning till the end of time. You can imagine this is mathematical task of such scale that it has to be ruled out as a rational possibility.

I have to concur with c0de here. Even in you own statement above, you've contradicted yourself. An all-knowing creator God would know all. What you are doing is setting limitations based on what we humans cannot measure. But if God is an Infinite Mind, why would you assume that he isn't capable.

Chaos seems to rely on unpredictability. But it only seems unpredicatable to us because we evidently aren't capable in measuring all the variables that are the sources of the cause of some event. Indeed, in quantum mechanics, our very observation of something appears to influence the result. That may be because we are within the system which we are attempting to observe. But if God is outside the space-time continuum, then His observation would have no effect on the result.
 
lol, weak, weak, weak reasoning. The viewable universe is big. VERY BIG!! Like it takes a light-speed photon over 13 billion years to reach us from the furthest we can see.So this guy who likes to see his son nailed to a cross, or his prophet nail a 9 year old, creates all this and makes it so chaotic just to confound the number crunchers :D Gimme a break!! No two things in the relative universe are identical. So "it" had to create each one individually just to please your superstition :rolleyes: Seems likely :rolleyes:

These arguments are absolutely meaningless. There are far better counters.

tao
 



@ Tao


The viewable universe is big. VERY BIG!! Like it takes a light-speed photon over 13 billion years to reach us from the furthest we can see.So this guy who likes to see his son nailed to a cross, or his prophet nail a 9 year old, creates all this and makes it so chaotic just to confound the number crunchers :D Gimme a break!! No two things in the relative universe are identical. So "it" had to create each one individually just to please your superstition :rolleyes: Seems likely :rolleyes:

... None of this is relevant to the objections made by Dondi and myself against your argument.
 
Back
Top