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FQXi FORUM
August 19, 2017

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: Children of the Cosmos by Sylvia Wenmackers [refresh]
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Author Sylvia Wenmackers wrote on Mar. 13, 2015 @ 21:07 GMT
Essay Abstract

Our mathematical models may appear unreasonably effective to us, but only if we forget to take into account who we are: we are the children of this Cosmos. We were born here and we know our way around the block, even if we do not always appreciate just how wonderful an achievement that is.

Author Bio

Sylvia Wenmackers is a professor in the philosophy of science at KU Leuven (Belgium). She studied theoretical physics and obtained a Ph.D. in Physics (2008) as well as in Philosophy (2011). In her current project, she explores the foundations of physics, with a special interest in infinitesimals and probabilities.

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Author Sylvia Wenmackers wrote on Mar. 13, 2015 @ 21:12 GMT
Full title

"Children of the Cosmos - Presenting a Toy Model of Science, with a Supporting Cast of Infinitesimals"

One sentence summary

As children of the Cosmos, we should remember this: "It is not nature, it is scientists that are simple." ;-)

Synopsis

- we are selected (2.1);

- our mathematics is selected (2.2);

- the application of mathematics has degrees of freedom beyond those internal to mathematics (2.4);

- and, still, effective applications of mathematics remain the exception rather than the rule (2.3).

Deleted scene

[Voice over for intro] On a particular planet in this Universe, a species evolved, the members of which had some crude tools for measuring and an organ for thinking. Using their tools, they were able to create somewhat less blunt tools. Soon, they thought themselves gods.

Goofs

Well, I forgot to include the original anecdote that got me thinking about this topic. [Trigger warning: armchair philosophy.] I attended a lecture in which the speaker claimed that "There is a matter of fact about how many people are in this room". Unbekownst to anyone else in that room, I was pregnant at the time, and I was unsure whether an unborn child. To me, examples like this show that we can apply mathematically crisp concepts (such as the counting numbers) to the world, but only because other concepts are vague (like 'person').

Trivia

This is the first FQXi contest that I participate in. My main goal was to compose an accessible piece, with the risk that it is too basic for the specialists on this forum.

Best wishes to all,

Sylvia Wenmackers



Gordon Watson replied on Mar. 14, 2015 @ 03:54 GMT
Sylvia,

Thank you for the lovely essay: not too basic for me; but then, I'm not too much a specialist. However: my subsequent re-readings have been punctured by my selection bias (toward maths, and it as the best logic). Heretofore largely subconscious, I thank you for the reminders (and they do you no harm).

The lead-in to your essay was another good one by Tom Phipps, "On...

view entire post


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James Lee Hoover wrote on Mar. 13, 2015 @ 23:35 GMT
Sylvia,

Quite interesting essay -- a timely argument in light of certain false discoveries like BICEP2. Selection bias I suppose is like confirmation bias which leans more toward science studies.

I love your selection of quotes by Einstein and Newton. Einstein's reminds me of the saying about democracy. In addition, the quotes are quite appropriate for your essay.

We might expect that peers of scientists, through peer review and their commitment to scientific truths would reveal fallacies, as they did BICEP2, and thus selection bias. Now confirmation bias in the political world and the polarized world doesn't have objective overseers on either side.

You might find my "Connection: mind, physics and math" of interest regarding what the union of mind, physics and math have accomplished.

Regards,

Jim

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Sophia Magnusdottir wrote on Mar. 14, 2015 @ 07:02 GMT
Hi Sylvia,

I like the way you address the question about the effectiveness of math, but I also think you miss the actual puzzle there.

Sure, we evolved by natural selection that would favor us being able to extract laws and regularities, so in a sense evolution worked towards a species that would end up using math to understand its environment. However, that doesn't explain why we find ourselves in an environment that displays such regularities to begin with. See, it is fairly easy to conceive of some mathematical structure that is just a complete mess, is chaotic, not causal, does not lend itself to a perturbative approximation and 2nd order differential equations with well-defined initial conditions etc. Why do we find ourselves in an environment in which that is the case? An environment that had these laws for us to discover?

Only reason I can come up with would be anthropic - quite possible life can't develop unless there are some regularities, in time or in space, ie self-similarities in some sense. But I don't think anybody knows how to make that argument precise.

In any case, I actually don't think math is all that efficient at all. Just look at all the things we cannot describe by math! Maybe you find some time to look at my essay, I think you would find we have some things in common, even though I pushed my argument into a different direction.

-- Sophia

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Gordon Watson replied on Mar. 14, 2015 @ 22:55 GMT
Dear Sophia (and Sylvia),

As a local realist, I'm keen to study and learn about the nature of reality. Thus, as I say in my essay, I seek to ensure that there is nothing relevant missing, and nothing irrelevant found, in the models that I develop. Hence the maxim: "Every relevant element of the subject physical reality has its counterpart in our analysis; and there are no...

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Georgina Woodward wrote on Mar. 15, 2015 @ 21:28 GMT
Hi Sylvia,

A very enjoyable read. I did find it accessible and thought provoking.

Re thinking the unthinkable- I think we can think about the unthinkable without actually being able to think it : ) If I look at a cup I see one viewpoint of it. However emanating from its surface is potential sensory data- that has the potential to give many different views. The whole truth of what it, the object, is would be like taking all of that data at once, not a tiny sub set, and forming an image. If an amalgamated manifestation is formed showing all viewpoints at once, the many different outputs would not allow clear definition of any singular form -too much information at once would cause the image to be a blur.

So while we can imagine viewpoints not seen individually we can not imagine all of them at once. The source of all potential manifestations, the object, is not altered by which manifestations of it are or are not fabricated. So the source object might be considered to be before and after observation in a superposition of all orientations, relative to all possible observers. Only when a manifestation is formed by an observer is it thought to be as it is seen -one viewpoint rather than all. This is a transition across a reality interface, the observers sensory system in this case, ( that transition corresponding to hypothetical wave function collapse ) from what is independent of observation to what is observed to be. Leaning not towards an abstract Platonic realm of perfect mathematical objects, that you mention, but a realm of concrete absolute source objects and complete information.

Good Luck , Georgina

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Dipak Kumar Bhunia wrote on Mar. 16, 2015 @ 12:08 GMT
Dear Dr. Wenmackers,

Thanks for "Children of Cosmos". You are rightly said regarding selected roles of such "Children of Cosmos" or "mathematizing mammals" where you have concluded:"we are selected" and "our mathematics is selected" as well.

But from where or how that selection came from? Or who made that selection? Probably you are not conjecture to mean to have direct intervention of GOD? Rather, if such a selection is a kind of logic or pattern (the term what you have used) followed through casual steps (in cosmological evolutions) onward arrow of time (as like as Darwinian steps in biological selections by same nature) would be understandable.

However, instead of such "mathematizing mammals" centric views about the nature (it may include universe or multiverse but makes no differences) to link the physics and mathematics, why one could not think the same nature as if a huge set of some intrinsic elements e.g. all hardwares (to deal with physics) plus all softwares (to deal with mathematics), and all "innate cognitive abilities" in those "Children of Cosmos" (to deal with neuro-cognitive cyber-sciences) are fundamentally emerged out from the combinations of such hardwares & softwares to explore that nature through all "Toy models of science" no matter how many times are modified onward cosmological arrow of time? Because such "Children of Cosmos" is also a part of that nature.

Thanks

Dipak Kumar Bhunia

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Member Rick Searle wrote on Mar. 18, 2015 @ 02:14 GMT
Dear Sylvia,

I loved your essay, and you have a wonderful writing style. In large part I agree with you- that our mathematical thinking has been selected for by nature as part of our ability to recognize patterns. I also agree with you that there are limits to this sort of mathematical thinking.

What I wonder though, is if mathematical "thinking" is selected for, not just in humans, but in animals- such as was shown to be the case in the "Honeycomb Conjecture" how can we say that it is not a property of the world that exists independent of humans?

The conclusion of your essay where you suggest we need to practice intellectual "judo" to see beyond mathematics to other ways of structuring the world I found especially intriguing. Do you have any hints as to what such thinking might entail?

Again, really enjoyed your essay. Please take the time to check out and vote on mine:

http://fqxi.org/community/forum/topic/2391

Best of luck in the contest and all of your endeavors,

Rick Searle

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Marcel-Marie LeBel wrote on Mar. 18, 2015 @ 02:54 GMT
Sylvia,

Wondering about the pieces of the puzzle, we tend to forget about the board; that which underlies all the pieces and allows them to exist in the first place.



“Could our cosmos have been different – so different that a mathematical description of it would have been fundamentally impossible” I don`t think so. We are the universe looking at itself; our logic, our maths are that of the universe.

Best of luck,

Marcel,

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Marcel-Marie LeBel wrote on Mar. 18, 2015 @ 02:54 GMT
Sylvia,

Wondering about the pieces of the puzzle, we tend to forget about the board; that which underlies all the pieces and allows them to exist in the first place.



“Could our cosmos have been different – so different that a mathematical description of it would have been fundamentally impossible” I don`t think so. We are the universe looking at itself; our logic, our...

view entire post


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Edward Michael MacKinnon wrote on Mar. 23, 2015 @ 04:49 GMT
Sylvia,

A very stimulating essay. I differ on the selection process. For 95% of human existence humans, like their hominid predecessors lived in small Hunter-Gatherer groups. Here Darwinian evolution was operative in a way that it is not now operative. Most individuals did not live long enough to reproduce. This inculcated the dispositions needed for survival, willingness to sacrifice for the sake of the group and some form of altruism within the group. Mathematics could develop only when a more sedentary life style developed. In the very different universe you consider I doubt if hominids could have survived.

Edward MacKinnon

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Sujatha Jagannathan wrote on Apr. 1, 2015 @ 11:10 GMT
In accordance with your work, I would like to get answers from you :

If MAN is born only because mathematical bound how the physics came in this picture?

If everything are here in numbers with no texture and color which inanimate and animate would live?

Regards,

Miss. Sujatha Jagannathan

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Harry Hamlin Ricker III wrote on Apr. 2, 2015 @ 13:23 GMT
Dear Sylvia, Good job on this essay. It is a better essay that the ones I have read so far. However, as I read, I became less and less convinced of you thesis and at the end felt unsatisfied. You are on the right track regarding the selection effect. In my opinion this issue is not complicated. We invent the universe as we imagine it to be. That process involves mathematical imagination. A lot of that is not effective as being persuasive in terms of explanation. Yet some of the imagination works well as in classical physics. But that is based upon infinitesimal calculus. As you show that is not correct mathematics. So we invent some new mathematics that fixes the problem. I think the real question ought to be how is it possible for wrong mathematics to be so effective in physical science. That is because as we learn we discover that the math used before is wrong in many ways. So we are constantly reinventing the math to fit the universe. I think this is because humans want a mathematical universe, not because the universe is mathematical.

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James Lee Hoover wrote on Apr. 14, 2015 @ 02:59 GMT
Sylvia,

Time grows short, so I am revisiting essays I’ve read to assure I’ve rated them. I find that I rated yours on 3/13, rating it as one I could immediately relate to. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.

Jim

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Author Sylvia Wenmackers replied on Apr. 22, 2015 @ 10:30 GMT
Dear Jim,

Thank you for both of your comments. I finally got round to reading and commenting on your essay as well.

Best wishes,

Sylvia




Sylvain Poirier wrote on Apr. 21, 2015 @ 05:55 GMT
Hello. In your essay you wondered how "to imagine a world that would defy our mathematical prowess", in fact you mean : that would be anything else than highly mathematical in the way it was found.

In my essay I gave some precise expressions of how remarkably mathematical the universe is, what does this precisely mean. Moreover I do hold that, aside the remarkably mathematical aspects,...

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Member David Hestenes wrote on Apr. 22, 2015 @ 01:22 GMT
Sylvia,

Here are a few points for you to consider.

There are patterns in nature, but only reproducible patterns can be modeled with mathematics.

When there are reproducible patterns that we cannot model with our mathematics, we invent new mathematics to do the job. That is where most math comes from and why it works so well.

You say that probability and statistics informs statistical mechanics. I submit that the opposite is closer to the truth. On this point see

"Probability Theory, the logic of science" by E.T. Jaynes

–– one of the great books of the twentieth century!

.....David

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Author Sylvia Wenmackers replied on Apr. 22, 2015 @ 09:09 GMT
Dear David Hestenes,

I am grateful for your comment.

I am very fond of the work of E.T. Jaynes! For instance, in my course on the philosophy of probability theory, I teach his analysis of the chord paradox. But I take your point: my statement about statistical mechanics was sloppy at best.

Best wishes,

Sylvia




Cristinel Stoica wrote on Apr. 22, 2015 @ 07:21 GMT
Dear Sylvia,

I enjoyed reading your essay. Indeed, we understand the world, at least as much as we do, because we are its children, and this may explain why our math is effective in our physics. The four elements, in particular selection, support very well your thesis, and the example of non-standard analysis is well chosen. The style is eloquent, pleasant and funny just as much as it should be (I loved the disclaimer). I am very glad I didn't miss your essay!

Best wishes,

Cristi Stoica

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Author Sylvia Wenmackers replied on Apr. 22, 2015 @ 09:11 GMT
Dear Christi,

Thank you for your positive comment!

I had already read your text, but waited for the deadline to submit all my comments in one batch: it is now in your forum.

Best wishes,

Sylvia




Michel Planat wrote on Apr. 22, 2015 @ 08:29 GMT
Oups: "it is not nature, it is scientists that are simple".

Thanks Sylvia, I agree with most you are writing, a truly Darwinian essay.

Michel

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Author Sylvia Wenmackers replied on Apr. 22, 2015 @ 10:00 GMT
Dear Michel,

Thanks for your kind comment. I also wrote a reply to yours.

Best wishes,

Sylvia




Michel Planat wrote on Apr. 22, 2015 @ 12:11 GMT
Dear Sylvia,

Your paper really needs more comments than I was able to deliver in such a short time left to us. I love your disclaimer. But I also enjoy concepts as the multiverse, the maxiverse, the megaverse, the babyverse, the monsterverse., everything chaotic, exotic, sporadic, anomalous probability distributions... With them it seems that we are are closer to the complexity of the world internal or external to us. Thank you for your read of my dialogue and your positive appreciation

Best wishes,

Michel

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William T. Parsons wrote on Apr. 22, 2015 @ 20:02 GMT
Hi Sylvia—

Your essay is superb! It is both creatively crafted and well argued. Moreover, I agree with both your main argument and your supporting elements. Of course, my admiration for your essay may thus be a product of mere “selection bias”.

Speaking of selection bias, I whole-heartedly subscribe to your point-of-view that we are blind to “ubiquitous failures” when assessing the efficacy of mathematics. Before reading your essay today, I had a very polite back-and-forth with Cristi Stoica on this very subject regarding his essay. I made similar comments on Lee Smolin’s threads. Amusingly, before settling on the essay that I posted, I had considered doing an essay for this contest that forthrightly addressed the many ways in which mathematics fails to efficaciously describe the physical world. I had tentatively entitled the piece, “On the unreasonable ineffectiveness of mathematics in the natural sciences”. Given Section 2.3 of your essay, I’m glad that I moved in another direction.

The only area in which we seem to disagree is on the subject of the “unthinkable”, especially with respect to randomness. For starters, I bristle at such phrases as “totally random” or “pure randomness”. That’s like saying that a man is “totally dead” or that a women has a “pure pregnancy”. Something (an event) is either random or it is not. Statistical distributions, such as the Gaussian, are an entirely different kettle of fish (which, I think, was the point you were trying to make).

Furthermore, I do not agree that “unthinkable” worlds are so unthinkable. Here’s one: A world of “white noise” in which all variables have an amplitude greater than your field-of-view. Sure, we can define “white noise” from the outside. But living it, on the inside, would be another matter. You’d probably be wiped out in the ensuing chaos before you could even voice the thought, “Wow, this world may be based on white … argh!’. Here’s another: Your “Daliesque” world, except one in which the “laws of nature” change randomly and drastically (and not as in, say, a Gaussian way with small-scale random effects) and do so at random times. There would be no meta-regularities. Here’s a third: A world in which there were no discernable “events” or “objects”; there’s just amorphous “stuff”.

We are fortunate to live in a Universe that consistently displays regularities. This enables us to engage in reliable pattern recognition and, hence, algorithmic compression. Which allows us to do mathematics. Which then allows us to do mathematical physics. What a beautiful selection effect!

For reasons that escape me, your essay is terribly under-rated. I shall now try to adjust that.

Very best regards,

Bill.

post approved


Author Sylvia Wenmackers wrote on Apr. 22, 2015 @ 21:06 GMT
Dear Bill,

Thank you for your detailed and constructive feedback! It is sort of reassuring that you considered similar ideas for your essay. And even nicer that -in the end- we didn't come to this party wearing an identical outfit. ;)

I do think that it makes sense to speak of "totally random" as opposed to "partially random". I use the term "totally random" in situations where there are equal probabilities pertaining to two or more possible outcomes and "partially random" when there are non-equal probabilities. On this view, pure randomness hits a strange equilibrium between knowledge and uncertainty: it does not represent total lack of knowledge (because then we wouldn't even know what the possible outcomes are*), yet it does represent maximal uncertainty regarding which of the possible outcomes will be realized at the next instance of the relevant process. Still, at the group level, we do know a lot about random events (both for total and partial randomness). I do think that this sense of total randomness is an idealization, and of course we can never demonstrate something to be totally random (or even falsify it: a fair coin may keep coming up heads for however long we try, it's just exceedingly unlikely).

[*: unless the randomness is implemented at a higher level, as in a world randomly switching between laws, as you proposed.]

Thank you for your vote: it made my day. :)

Best wishes,

Sylvia




Member Marc Séguin wrote on Jun. 9, 2015 @ 19:13 GMT
Dear Sylvia,

I read your essay with great interest when it was posted, but I didn’t comment on it while the contest was underway: I had read the disclaimer at the end of your essay, “No parallel universes were postulated during the writing of this essay”, and since my own essay postulates an infinite ensemble of parallel universes and multiverses, I preferred to keep a low profile!...

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Author Sylvia Wenmackers replied on Jun. 11, 2015 @ 13:29 GMT
Dear Marc,

Thank you for your detailed and kind reply.

The distinction you make here, between mathematics and Mathematics, is really helpful for these kinds of discussions. (It would even have been a great starting point for an essay!) When we apply some of the idealizations that go on in mathematics to the field itself, we obtain the concept of Mathematics. This move has been made at least since Plato, and seems to come so natural to us, that it often goes unnoticed. So, it is very helpful to indicate when this is going on. Indeed, I tried to stick to mathematics in the real world, because it is not clear to me that 'Mathematics' refers to anything other than the human concept thereof. But I certainly don't mind to speculate about what it implies if Mathematics would have an indepent existence. So, I will also post a reaction to your essay on your forum.

Best wishes,

Sylvia




Christine Cordula Dantas wrote on Jun. 11, 2015 @ 10:11 GMT
Congratulations and best wishes!

Christine.

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Author Sylvia Wenmackers replied on Jun. 11, 2015 @ 13:32 GMT
Dear Christine,

Thank you. I remember reading and commenting on your essay. As I wrote then, I quite liked it, so I am not surprised that it got you a prize, too. Congratulations. :)

Best wishes,

Sylvia




Georgina Woodward wrote on Jun. 12, 2015 @ 01:35 GMT
Dear Silvia Wenmackers,

congratulations on your prize.

I'm sorry that I forgot to say what a good essay. I was preoccupied with trying to make the point Re thinking the unthinkable- we can think about the unthinkable without actually being able to think it. That is probably more interesting to me that to you, as it received no response. Though I am pleased to see that you did respond to some comments by other people.

Well done, enjoy your prize, kind regards Georgina

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Georgina Woodward replied on Jun. 12, 2015 @ 05:33 GMT
This may be interesting to you. Michael Hansmeyer talking about building shapes that can not be imagined because they are too complex and at a scale of folding that can't be carried out by human beings. He shows that how to produce these shapes can be thought about quite simply, even though the output itself is unimaginable. It is a very simple process likened to morphogenesis and he mentions breeding of types to produce new designs. He also mentions designing processes rather than shapes in the future. So it seems we are not limited by our imagination. Michael Hansmeyer: Building unimaginable shapes, kind regards Georgina

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Author Sylvia Wenmackers replied on Jun. 12, 2015 @ 11:57 GMT
Dear Georgina,

Thank you for messages.

I did read your earlier message and did find it an interesting point of view. Since you didn't ask any questions (and - as a finite being - I am always short on time) I did not answer back then. ;)

Your comments remind me of fractals: at the same time, easy to define and impossible to image in all their intricacies. It also reminds me of so-called 'intangible objects' of which no explicit example can be given (because their existence relies crucially on a non-constructive axiom). For instance a free ultrafilter, which is crucial for the infinitesimals that I mention in my essay.

In one way of seeing it, such objects cannot be imagined, but in another way they can be: I can understand the definition of a free ultrafilter and understand the existence proof, and at the same time understand that no explicit example can be given. If there is at least one viewpoint in which an object can be imagined (in this case via the definition), I consider it to be 'imaginable': this definition was developed by people and in that sense imagined, even though we do not necessarily grasp all the consequences. I do not require this kind of transparancy or omniscience for using the word 'imaginable'.

Thank you for the link to the TED-talk; I will watch it this evening. :)

Best wishes,

Sylvia




Eckard Blumschein wrote on Jun. 13, 2015 @ 11:25 GMT
Dear Silvia Wenmackers,

Neither the title nor the abstract of your essay sparked my interest. I was surprised how clever you managed to precisely tackle the topic in your essay and just to ignore pointless skeptical comments.

However, I wonder if you are in position to substantiate what you wrote on p. 6: "NSA [you referred to Non Standard Analysis, not to Natural Science Alliance, not to National Security Agency, ;-)) ] seems a very appealing framework for theoretical physics: it respects how physicists are already thinking of derivatives, differential equations, series expansions, and the like, and it is fully rigorous."

Appealing to you might not be enough to me. I am claiming that there are compelling reasons to abandon putatively rigorous but actually unwarranted naive set theory. That's why I consider Abraham Robinso(h)n not even wrong when he wrote: "any mention, or purported mention, of infinite totalities is, literaly, meaningless". Isn't NSA meaningless?

Eckard Blumschein

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Author Sylvia Wenmackers replied on Jun. 15, 2015 @ 13:17 GMT
Dear Eckard Blumschein,

Thank you for your comment and question.

The main point of 2.4 is the observation that in applying mathematics to something non-mathematical, there are many degrees of freedom. This point applies even if one would reject some parts of mathematics.

I regard almost all of mathematical statements as expressing suppositional knowledge: assuming these axioms, this follows; assumming those axioms, that follows; etc. (In practice, the first part is often silent and fixed by context only.) In that sense, one does not need to accept any set of axioms, merely check what follows from them. The resulting suppositional statement can be meaningful even if one is not willing to accept any of the axioms. So, I would not say that NSA is meaningless, rather that both standard and non-standard analysis are 'useful fictions' that do produce suppositional knowledge.

Best wishes,

Sylvia




Steve Dufourny wrote on Sep. 3, 2015 @ 09:49 GMT
Hello dear Ms Wenmackers

Congratulations for your essay and your humility Inside this universal sphere in spherisation. We are just indeed simple and we continue a humble road.

The philosophy shows the road of this universal love in improvement, spherisation for me.Indeed when we understand what we want, it is simple to understand that we are just Young annimals inteeligent in evolution spherisation.I am asking me where the humility is encoded in our quantum sphères :)

congratulations for your work , it is relevant

Best Regards

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Lee Bloomquist wrote on Apr. 16, 2016 @ 01:41 GMT
Sylvia, on NSA--

properTime = (clockTime, properTime)

and

clockTime = (nonstandardPast, standardPresent, nonstandardFuture)

There is a bi-conditional suggesting an exchange of information between NSPast and NSFuture,

thus generating a Born infomorphism--- all made possible by the simplest possible application of NSA.

Can you suggest a next step?

http://fqxi.org/community/forum/topic/2420

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Lee Bloomquist wrote on Sep. 19, 2016 @ 09:47 GMT
Sylvia,

Sorry, way too general. Here is a more specific question—

There is a "logical black hole" in the standard analysis of the Standard Model of Particle Physics.

Do you think it might be worthwhile to model this logical black hole as a physical black hole?

The logical black hole is implicit on page 2 of Abraham Robinson's book "Non-standard Analysis," where he wrote:

"To this question we may expect the answer that our definition may be simpler in appearance but unfortunately it is also meaningless."

In the previous paragraphs, Robinson had translated an epsilon-delta definition from standard analysis into a meaningful definition using non-standard numbers.

Like a physical black hole, his translation took us inside of H. Jerome Keisler's "infinite microscope." But once inside we cannot get back out, because of the above quote.

Although statements about very small real numbers like epsilon and delta might be meaningfully translated (as Robinson did) into statements about non-standard infinitesimals, the statements in question about non-standard infinitesimals cannot be meaningfully translated into statements in standard analysis about real numbers like epsilon and delta.

It is a "logical black hole."

Is it physical?

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