Grzegorz Kołczan, the author of "Aristotle principles of dynamics", suggested to me that the discussion on this blog is not well organized for a serious discussion. Therefore I decided to devote a separate post to this subject. Here are the collected comments from the previous post, related to the subject.
Mathilde S (forwarded by A.J.): July 16, 2023 at 3:21 PM
" ... regarding Aristotle, I appreciate that he created a certain current in metaphysics, nevertheless Aristotle is not an idealist philosopher, so I much prefer the philosophy of Plato, Plotinus or Hegel.
At the same time, I believe that physics is essentially Platonic and should be developed in that direction, whereas I consider the introduction of Aristotelianism to be pointless. Physics constructed in this way will go no further for a number of reasons that I could discuss ...."
Mathilde S.July 16, 2023 at 4:19 PM
Since I have been called to the board, I will be more specific. With questions from other commentators or the author, I can gradually make my statement more detailed.
There are a few philosophic and scientific arguments that could support the idea that physics is essentially Platonic and should continue to be developed in that direction, while an Aristotelian approach might be limited.
One of them is the mathematical structure of the Universe. This is a strong argument for a Platonic viewpoint in physics. Many physicists, including Max Tegmark with his Mathematical Universe Hypothesis, argue that the universe is not just described by mathematics, but it is a mathematical structure. This idea reflects Plato's Theory of Forms, which posits a world of perfect, unchanging ideals beyond our sensory experience.
The next one is connected to physical laws and universality. The laws of physics as we know them, like gravity and electromagnetism, are unchanging and universal. They apply regardless of time or space. This is similar to Plato's Forms, which are timeless and universal. If the ultimate laws of physics exist in a similar Platonic realm, this could provide the theoretical unity that many physicists seek.
Moreover, quantum mechanics and related theories have been interpreted in ways that are favorable to a Platonic interpretation. For instance, the many-worlds interpretation of quantum mechanics posits an infinite number of parallel universes to explain quantum phenomena. This seems more in line with the Platonic metaphysical idealism than with Aristotelian material realism.
Now let's consider a few reasons why you might view Aristotelian approaches as limited:
Aristotelian physics is largely empirical, grounded in what we can observe and experience. While this has led to the development of the scientific method and the accumulation of knowledge, it does not necessarily help in areas where direct observation is not possible, such as quantum physics or cosmology.
At the same time Aristotle's approach is less amenable to abstract mathematical reasoning, which is essential to modern physics. While Aristotle believed in the use of logic and reasoning, he focused more on concrete, observable phenomena, rather than the abstract entities that dominate modern physics, such as fields, forces, and particles.
Of course, this is a simplification of both Platonic and Aristotelian philosophies, and their applicability to physics. There are aspects of each that can be beneficially incorporated into scientific thought, as both have already influenced the development of physics in profound ways.
Nevertheless, physics is becoming more and more abstract and, consequently, should be developed more in the direction of Platonism. This does not change the fact that in some specific cases Aristotle's approach can be useful, but I think that these are only concrete models, not theories that will actually significantly influence the development of the so-called 'new physics'. This subject has also been addressed by Michał Heller in many books and publications, quoting numerous arguments for the Platonic approach.
Mathilde S.July 16, 2023 at 4:22 PM
At the same time, for Polish speakers and those interested in this topic, I can also recommend a lecture by Krzysztof Meissner.
I do not like all of Meissner's statements, but I nevertheless agree with him on the core issue - Platonism.Grzegorz Kołczan Forwarded by A. J. July 16, 2023 at 5:11 PM
G.K. replied to me in a private message as he is having trouble seeing comments on his PC:
"I, for one, can't get into the discussion now, as I am finishing editing the monograph. However, I see that there is a dispute emerging on the Plato-Aristotle line.
I may know Plato poorly, but Aristotle is undoubtedly closer to physics - and not just ancient physics, but also quantum physics, as Rovelli shows. The infatuation of physicists with Plato, I believe, is a misunderstanding of what "infatuation" with one idea is. Aristotle trumps Plato in every scientific respect, except one: Aristotle did not have an outstanding student
Mathilde S.July 16, 2023 at 5:36 PM
"I, for one, can't get into the discussion now, as I am finishing editing the monograph. However, I see that there is a dispute emerging on the Plato-Aristotle line.
I may know Plato poorly, but Aristotle is undoubtedly closer to physics - and not just ancient physics, but also quantum physics, as Rovelli shows. The infatuation of physicists with Plato, I believe, is a misunderstanding of what "infatuation" with one idea is. Aristotle trumps Plato in every scientific respect, except one: Aristotle did not have an outstanding student ? .".
I am in a position to undertake an argument, so if any specific arguments are made I am happy to discuss.
For the moment I would certainly agree that Aristotle is closer to classical mechanics. I am not going to debate here, I think we both agree.
As for quantum mechanics, on the other hand, I would need to see the arguments. In terms of what I think physics should aim for, I'm unlikely to be persuaded by Aristotle's approach, but I'm curious to try to convince me.
And what is the infatuation with one idea? Well, physics was largely born out of aesthetics. I am very curious about this discussion.
Mathilde S.July 16, 2023 at 5:44 PM
I have one more question: what is the position of the blog author? Platonism or Aristotelianism?
Arkadiusz JadczykJuly 16, 2023 at 5:47 PM
Already answered:
Quotation from Daigneaults paper (Ref. [4]):
"Genesis teaches that : « …et Dieu dit: ‘ Que la lumière soit ! Et la lumière fut. » ; « and God said : Let there be light ! And light there was ».
Some say that at MIT, Caltech, and other trade schools, one often sees engineers wearing T-shirts that display Maxwell's equations as God's real words just before there was light
[ http://wiki.yak.net/591/howto.pdf].
Maybe this is so for the following facts which do not seem to move astrophysicists should not leave any mathematician indifferent especially those like myself who believe that God is Himself a mathematician ! " (A. Daigneault, Irving Segal's Axiomatization of Spacetime and its Cosmological Consequences)
The "Aristotle" discussed here is certainly a different than usual interpretation of Aristotle. Most people would not see atomism as an extension from Aristotle. Going from atomism to a pluralistic idealist Leibniz monad might not be difficult. Aristotle did have the elements and thus via Ark's wife Laura, information. I got to physics math via personality models where the elements have been rather directly used as personality factors. Clifford algebra is rather directly information with its 2^n dimensions. You don't work with very high n's either. Even if you go up to n=8, you work a lot with 4s, 5s and 6s as Ark has shown recently.
Not getting too far away from classical is perhaps a common problem for the quantum researchers. One of the main points of EEQT is to get some classical into the quantum. I didn't understand EEQT in even my usual not very detailed math way until I added some general understanding of differential geometry which has classical origins. Feynman paths are perhaps a more classical visual for quantum than the wave function.
Hello everyone,
ReplyDeleteis a test message sent from a smartphone.
Grzegorz M. Koczan
Comments are moderated (by me). So sometimes they show up immediately, sometimes after hours, sometimes never (when inappropriate or private).
ReplyDeleteStrona 24 poniżej (115):
ReplyDeleteLet p and q are two points in N ->
Let p and q are two points in PV
@ Bjab I have moved your comment to the previous post since the pdf file it is concerned with is displayed there and not here.
ReplyDeleteTyping somehow works, so I'm writing a short post.
ReplyDelete@ Mathilde S,
Thanks for starting an interesting topic of discussion. I understand that the discussion concerns Plato and Aristotle in the context of the sciences: (i) mathematics, (ii) physics, (iii) quantum mechanics. The last one is separated in a historical context.
I understand that my arguments for classical physics (in the description of the principles of motion) for Aristotle are accepted. Yes, I didn't make an argument for quantum mechanics in the context of Aristotle, but I didn't see anyone make one for Plato either (@ M.S.).
We'll get to it.
First, however, I propose to answer the following question:
Name one important mathematical discovery of Plato?
It cannot be the foundation of the academy, it cannot be the Platonic solids, it cannot be Theaetetus, it cannot be the discovery of irrational numbers.
@ John G, A. Jadczyk,
I am also asking you to answer the same question. I assume you're more for Platonism - if you want to change to Aristotle, you don't have to answer :-).
We'll do other things later.
Grzegorz M. Koczan
"I understand that the discussion concerns Plato and Aristotle in the context of the sciences: (i) mathematics, (ii) physics, (iii) quantum mechanics."
DeleteThere is a logical confusion in the above sentence. The term "sciences" has a wider meaning than those three listed. It includes also a general philosophy and methodology of science: what science is about? How much can we know? And what is "knowledge"? Thus if we want to compare these two philosophers with open minds, it is necessary to extend the arena of the discussion.
We could also discuss in a narrow range, but then we can make narrower and narrower, and by doing so in a clever way, we will get to any conclusion that we have chosen before.
I think that a pragmatic approach can be useful here: what was the real influence of Plato and Aristotle on science till now? And what would be the prognosis for the future that is yet to come?
@AND. Jadczyk,
ReplyDeleteI am afraid that there can be no logical error in the selection of scientific disciplines for discussion, preceded by a colon.
Such a choice of disciplines corresponds to the post of Mathilda S.
There is no point in talking about everything and nothing. The topic can be expanded later, but you have to focus on something.
Grzegorz M. Koczan
Did I say "logical error"? No. I said "logical confusion". No, confusing "error" with "confusion" IS an error.
DeleteOn the other hand we can as well concentrate on "physics". But physics would not exist (except in a primitive form) without mathematics. And would not exist without physicists, who follow their intuitions. So, we need to discuss intuition as well.
Do we agree on this?
OK. Julian Barbour, a British physicist, in his book "The End of Time" has a whole chapter: Chapter 7: Paths in Platonia.
ReplyDeleteQuoting from http://www.voting.ukscientists.com/barbour0.html
"Platonia is a timeless jumble of practically infinite possibilities. The images most likely to come together in an appearance of time's motion are the kinematic slides best matched to each other.
If you cut a movie reel into its individual slides, you could put them back together in sequence by comparing those which were best matched to run continuously. In platonia, tho, there is an embarrassment of choice from every conceivable possibility of image, tho the vast majority are so ill-matched that they are easily dismissed from any probable historical sequence.
The explanatory success of quantum theory has made physicists take seriously the notion that there is a graded potential for all logical possibilities of existence to become reality. Barbour makes the point that these possibilities could include unimaginable heaven, purgatory and hell. ( He has a pantheistic belief in platonia, rather than in a personal god. )"
Do we have a point for Plato?
For me, Barbour's potentialities are Aristotle's potentialities.
ReplyDeletePlato still has 0 mathematical examples :-).
Quoting from
DeletePlato’s View on the Importance of Mathematics :
"Plato’s contributions to mathematics were focused on the foundations of mathematics. He discussed the importance of examining the hypotheses of mathematics. He also drew attention toward the importance of making mathematical definitions clear and precise as these definitions are fundamental entities in mathematics. Another indirect contribution of Plato was the important role he played in encouraging and inspiring people to study mathematics. In the Academy, he proposed many mathematical problems and encouraged the students of the Academy to investigate. This led to the appearance of many mathematicians, like Eudoxus, who contributed in the progress of mathematics. "
Do we have a point for Plato? Or should I look for more detailed and better documented sources?
Now a big point for Aristotle:
Delete"Aristotle is known for inventing the scientific method of analysis, which can be applied to multiple fields of study. He also is responsible for breaking fields of knowledge into categories and subcategories, such as psychology, biology, politics, logic, chemistry, and botany. His contributions to the different facets of theoretical study can be broken into three different groups known as speculative philosophy, natural philosophy, and practical philosophy. These are explained in depth below."
Big point according to my suggestion to extend the scope of the discussion towards philosophy of science in general.
Still modest for Plato - a mathematician in opposition to Aristotle - supposedly anti-mathematics.
ReplyDeleteIf Plato introduced definitions or axioms, I can consider that as a contribution to mathematics. However, I need the original (?) source.
A good source is David Bowler, "The Mathematics of Plato's Academy. A New Reconstruction", Clarendon Press Oxford 1999
DeleteFrom the Wikipedia article on Aristotelian physics: "Unlike the eternal and unchanging celestial aether, each of the four terrestrial elements are capable of changing into either of the two elements they share a property with... The elements are not proper substances in Aristotelian theory (or the modern sense of the word). Instead, they are abstractions used to explain the varying natures and behaviors of actual materials in terms of ratios between them."
ReplyDeletePlato had his forms which is very abstract information-like but Aristotle did the same thing as Plato with elements and even had the changing-unchanging aspect when aether gets added to elements. Elements using wet, cold, etc. attributes for change is even like fermions using bosons for change of state in an unchanging Platonia of all states. They both kind of had nice thoughts especially given how long ago they lived. Obviously more recent people have the advantage of a more knowledgeable starting point.
And on a somewhat lighter note:
ReplyDeleteAristotle quotes
Everybody will find there a personal pearl! And probably more than just one.
What is needed is Plato's quote from Plato's work, not the products of Plato's Academy.
ReplyDeleteOne example from Fowler's book:
Delete"2 . 3 ANT H Y P H A I R E T I C A L G O R I T H M S
2.3(a) The Parmenides proposition
Plato gives, at Parmenides 154b-d, a special case of a proposition that we shall use:
If one thing actually is older than another, it cannot be becoming older still, nor the younger younger still, by any more than their original difference in age, for if equals
be added to unequals, the difference that results, in time or any other magnitude, will always be the same as the original difference . . . . [But] if an equal time is added
to a gre ater time and to a less, the greater will exce ed the less by a smaller part (morion).
To paraphrase and simplify slightly Plato's homely paradox: as two people grow older together, the difference between their ages will remain constant, but the ratio of their ages will get closer to 1 : 1 ."
Should I quote more similar examples? Do they count?
@Grzegorz M. Koczan
ReplyDeleteWhile Plato is known more for his philosophy than his contributions to mathematics, he did have a significant influence on the development of mathematical thought. Your question, however, rules out most of his well-known indirect contributions like the Platonic solids or the fostering of mathematical thought through his Academy.
One important contribution that Plato made, which might be overlooked, is his mathematical philosophy that had a profound influence on the development of mathematical ideas. Plato proposed the idea that numbers and geometric figures exist as ideal forms, independent of the material world. This concept is often encapsulated in the phrase "the theory of forms." This belief that numbers and mathematical concepts were "real" in some sense helped support the development of abstract mathematical thought.
Plato also promoted the idea that the study of mathematics was a form of mental discipline that could lead to clearer thinking and better reasoning skills. This idea has continued to influence education systems throughout history and into the present day.
However, if you're looking for a specific mathematical theorem or discovery attributed to Plato himself in the same way as, for example, the Pythagorean theorem is attributed to Pythagoras, there is no such widely recognized discovery. His contributions were more indirect, through his influence on his students and his promotion of mathematical study.
When I speak of Platonism, I mean first and foremost that ideas really exist and the very degrees of existence referred to by Arkadiusz Jadczyk in the context of Julian Barbour's approach.
At the same time, as I mentioned, I am currently working on some mathematical approaches to Plato's philosophy, extracting them from his treatises. I will admit, however, that I am also paying attention to Plotinus and Hegel, but both of these philosophers are idealists. Plotinus' philosophy is a development of Plato's philosophy, Hegel's philosophy introduces an additional dynamic that can be interestingly captured in the terms of category theory.
In this context, Plato is for me like a simple column, Plotinus is a column with ornaments, while Hegel's philosophy illustrates something like the transformation of this column with ornaments. This is very difficult to describe with philosophy alone, hence my work on category theory.
I conclude that a well-defined metaphysics should underlie every physical theory. This is a very difficult task, but it seems to me to be a kind of original way to develop new physical ideas that are a consequence of this metaphysics.
@Grzegorz M. Koczan
ReplyDelete"For me, Barbour's potentialities are Aristotle's potentialities.".
I find this an interesting and non-obvious observation. Could you elaborate on this statement?
I just asked you if you want more examples? Your opinion that my exampke is "trivial" is just an opinion. You did not give any support. It is in general advisable to provide support for personal opinion. Now, Fowler was a professional mathematician and a historian of mathematics. So his personal opinion about importance of Plato for mathematics naturally counts for me more than a personal opinion of a physicist. But Fowler gives us not only personal opinion, but gives us a discussion and refers to Plato's original writings. What else do we need? More examples?
ReplyDeleteAnd concerning your qulification that something is trivial. I would advise avoiding such statements in similar disputes concerning the history of science.. Today the statement that the Earth is round and not flat seems trivial to probably most of the people on this planet. But it was not always so.
ReplyDelete@Grzegorz M. Koczan
ReplyDelete"Regarding Aristotle-potentiality-Barbour, I just meant only that the concept of potentiality was invented by Aristotle.".
While there isn't a direct equivalence to Aristotle's concept of "potentiality" in Platonic or Neoplatonic philosophy, the Platonic Forms and the Plotinian Intellect might be seen as analogous. They are the perfect and ideal realms, which contain the potential for all things in the physical world. In this way, rather than individual objects having potentiality, the entire physical world has potentiality emanating from the realm of Forms or the Intellect.
So, in essence, Plato and Plotinus see the potential for change and transformation not in the individual entities of the physical world, as Aristotle did, but in the metaphysical realms of Forms and the Intellect, from which the physical world emanates and towards which it strives.
"There is still no citation for introducing axioms and definitions.".
Plato’s philosophy had a profound influence on how mathematics was viewed and practiced, particularly in terms of the philosophical understanding of mathematical objects. Plato's belief in abstract, perfect forms laid a philosophical groundwork for the development of mathematics as a field of abstract thought. His philosophy insisted that mathematical objects and relations exist independently of the physical world, and this perspective contributed significantly to how mathematics was pursued in the Western world.
It seems to me that axioms and definitions are not everything. Plato writes about mathematics in a different way. His works are worth analysing using the historical-critical method. Are you familiar with the historian of philosophy Giovanni Reale? I don't agree with his views on Christianity, but I think he gives an interesting account of some aspects of Plato's philosophy. At the same time, in one of the files I uploaded, I write about understanding the dialogue 'Timaeus' in the context of modern mathematical analysis.
@ A. Jadczyk,
ReplyDelete"2 . 3 ANT H Y P H A I R E T I C A L G O R I T H M S
2.3(a) The Parmenides proposition
Plato gives, at Parmenides 154b-d, a special case of a proposition that we shall use:
If one thing actually is older than another, it cannot be becoming older still, nor the younger younger still, by any more than their original difference in age, for if equals
be added to unequals, the difference that results, in time or any other magnitude, will always be the same as the original difference . . . . [But] if an equal time is added
to a gre ater time and to a less, the greater will exce ed the less by a smaller part (morion).
To paraphrase and simplify slightly Plato's homely paradox: as two people grow older together, the difference between their ages will remain constant, but the ratio of their ages will get closer to 1 : 1 ." Fowler example of Plato mathematics tasc
"I just asked you if you want more examples? Your opinion that my exampke is "trivial" is just an opinion. You did not give any support. It is in general advisable to provide support for personal opinion. Now, Fowler was a professional mathematician and a historian of mathematics. So his personal opinion about importance of Plato for mathematics naturally counts for me more than a personal opinion of a physicist. But Fowler gives us not only personal opinion, but gives us a discussion and refers to Plato's original writings. What else do we need? More examples?" A. Jadczyk
"And concerning your qulification that something is trivial. I would advise avoiding such statements in similar disputes concerning the history of science.. Today the statement that the Earth is round and not flat seems trivial to probably most of the people on this planet. But it was not always so." A. Jadczyk
I see some abuses and ad personam arguments here.
1) Fowler does not call the example given Plato's great mathematical achievement.
2) "A horse is what everyone sees." Anyone can comment on the meaning of Fowler's example (Mathilda S, John G, and even A. Jadczyk). I have already said that the example is trivial - and there is nothing wrong with it, nothing inappropriate and nothing untrue. It's a rational opinion.
3) The example was supposed to be one, but strong and indisputable. Preferably being a quote from Plato, because a lot is claimed about what Plato did, and there are no sources for it.
"The example was supposed to be one, but strong and indisputable."
DeleteMy rational opinion is that there are no undisputable examples. Every example can be disputed concerning its impact. Scientists dispute everything. And it is good.
"Fowler does not call the example given Plato's great mathematical achievement."
Incorrect logic here. Did I ever say that Fowler called it a great mathematical achievement? No I did not say so. So it is a logical twist. It is my rational opinion and it is not ad persona argument. I am not saying that YOU are this or that. I am saying that this particular statement is disturbing to me. Do you see the difference?
@ G.K.
DeleteYou wrote: "I have already said that the example is trivial - and there is nothing wrong with it,"
But there IS something wrong with this. See my comment July 17, 2023 at 9:58 PM
@Grzegorz M. Koczan
ReplyDelete"A horse is what everyone sees." Anyone can comment on the meaning of Fowler's example (Mathilda S, John G, and even A. Jadczyk). I have already said that the example is trivial - and there is nothing wrong with it, nothing inappropriate and nothing untrue. It's a rational opinion.".
It seems to me that the subject of this discussion is not very clearly defined. I don't know about the author and other readers, but I don't intend to argue about whether Plato was in fact a great mathematician as we understand being a great mathematician today.
Instead, I believe that Plato's greatness was not about specific axioms, definitions or calculations (although I encourage you to look at my file on comparative analysis of Plato, Plotinus and Hegel, where I give an analytical argument about the dialogue 'Timaeus').
However, in adopting a Platonic approach, I am not guided by whether or not Plato actually did anything great in mathematics. What I have in mind here is his philosophy of mathematics rather than specific calculations. Calculations are secondary, they are done in some paradigm, having adopted some system of axioms.
@Arkadiusz
ReplyDelete"@ G.K.
"You wrote: "I have already said that the example is trivial - and there is nothing wrong with it,"".
However, even assuming that this example is trivial, I believe that this state of affairs does not disprove the power of mathematical Platonism.
"if you really sincerely believe that the statement about the constancy of the absolute difference in age and the decrease of the relative difference is Plato's great mathematical discovery "
ReplyDeleteDid I say so?
"You judged the quality of the argument by the education of the authors"
It is not that education counts but the expertise. A person without education may well become a top class expert in a certain area owing to his/her own deep studies of the subject. Concerning Fowler, he had both education and expertise.
Did we forget Aristotle? Quoting from Anagnostpoulos, "A Companion to Aristotle", 17. Change and its Relation to Actuality and Potentiality":
ReplyDelete" When Aristotle says that change is the actuality of what is potentially F, qua potentially F, the point of the qua clause is partly to emphasize that the actuality in question is the actuality of something insofar as it is merely potentially F. A thing’s potential to be F is most fully actual as a potential, when the thing is not yet F. "
That is what quantum theory is about. Potential becomes actual. From infinite possibilities to a selected actuality. How is it being selected? That is the problem yet to be solved. EEQT proposes (temporarily) piecewise deterministic algorithm (including RANDOM jumps).
@ G.K.
ReplyDeleteM.S. asked about potentialities in her comment :
`` "For me, Barbour's potentialities are Aristotle's potentialities.".
I find this an interesting and non-obvious observation. Could you elaborate on this statement? ''
@ Mathilda S,
ReplyDeletethank you for assuming that the Plato/Fowler/Jadczyk example is trivial. Agreed that if it is trivial, no further conclusions should be drawn from it. A better example should be found instead of claiming, like A.J., that small is big and that my judgment is of little weight to that of Fowler, who is not even among the actual debaters.
My question in the discussion is quite specific. Of course it is inconvenient and troublesome.
@ A. Jadczyk,
the question of an example of Plato's mathematicality is not finished yet. John G has yet to comment on this. Mathilda S found no example. And the example of A.J. is weak (in the opinion of G.M.K. and M.S.). I posted the source on ResearchGate, but no one has researched it here yet.
I firmly believe that the "argumentum ad personam" was made here - and denying this fact does not bode well for the discussion. No one even suggests asking if Aristotle has any mathematical example.
Quoting from : Plato’s Parmenides
Delete"The Parmenides inspired the metaphysical and mystical theories of the later Neoplatonists (notably Plotinus and later, Proclus), who saw in the Deductions the key to the hierarchical ontological structure of the universe.
1. Overview of the Dialogue
2. The Introductory Section: Zeno’s Argument 126a–128e
3. Socrates’ Speech: The Theory of Forms 128e–130a
4. Problems for the Theory of Forms 130a–134e
4.1 The Extent of the Forms 130a–e
4.2 The Whole-Part Dilemma 130e–131e
4.3 The Third Man Argument 132a–b
4.4 Forms as Thoughts 132b–c
4.5 The Likeness Regress 132c–133a
4.6 The Greatest Difficulty 133a–134e
5. How to Save the Forms: The Plan of the Deductions 134e–137c
6. The Deductions 137c–166c
6.1 The First Deduction 137c–142a
6.2 The Second Deduction 142b–155e
6.3 The Appendix to the First Two Deductions 155e–157b
6.4 The Third Deduction 157b–159b
6.5 The Fourth Deduction 159b–160b
6.6 The Fifth Deduction 160b–163b
6.7 The Sixth Deduction 163b–164b
6.8 The Seventh Deduction 164b–165e
6.9 The Eighth Deduction 165e–166c
7. Conclusion
Bibliography
Academic Tools
Other Internet Resources
Related Entries
"the question of an example of Plato's mathematicality is not finished yet. John G has yet to comment on this"
DeleteMy one previous post here in the comment section was meant to address that. Given these guys lived so long ago and thus had a much less knowledgeable starting point, I don't require very exact details for a discovery. Thus Plato's forms is fine for me as a discovery related to the math of information theory.
Also as I mentioned previously, Aristotle's treatment of the elements also is fine for me as a discovery for information theory. If Aristotle had added something like the actualities in the unchanging aether represent potentialities for the elements' changes I would have liked Aristotle even more.
@Grzegorz
ReplyDelete"Mathilda S found no example.".
Please read the file "A comparative analysis of the metaphysics of Plato, Plotinus and Hegel". There I give examples concerning the dialogue "Timaeus". You can download the file here: https://we.tl/t-yeBbwyQJsY
At the same time, I do not claim that my example is some very strong one. I believe it is not, but again I repeat that this is not about examples of specific axioms, definitions or calculations. What I am concerned with in Platonism is the approach itself. I am primarily referring to idealism. Even more than Plato I like Plotinus and Hegel and their philosophical systems, but they are a continuation of Plato's thought.
@G.K.
ReplyDeleteConcerning Vardulakis paper that you quoted, it is not about a false theorem of Plato, but by someone's else conjecture:
"This led him to believe that in this passage of the Laws,
Plato is in fact stating (in a cryptic way) a theorem, which
can be formulated as follows:"
@G.K. "Of course it was an "argumentum ad personam""
ReplyDeleteNo, it wasn't. It is my duty here to point out twists of logic in a discussion, when I spot them. I am not saying that I am free of such. Errare humanum est.
@Grzegorz M. Koczan
ReplyDelete"Name one important mathematical discovery of Plato?"
-------------------
Plato was not a mathematician but a philosopher.
His significance for mathematics lies not
in working in the area of mathematics but in the area of
philosophy - which allows mathematicians to
interpret the objects of their work as
eternally existing - of divine nature - and only discovered
by mathematicians in contrast to the impermanent constructs
created by humans. There are also mathematicians who are
adherents of a different philosophy, i.e. constructivists/intuitionists.
Claiming to point to a mathematical discovery
made by Plato seems downright bizarre.
Especially since Plato hid his knowledge from the
unworthy - according to the teachings learnt in Egypt ( Hermetism).
In mathematics one does not work using metaphors as Plato did-.
the opposite - everything must be concrete strict and pprecise.
"Dobry matematyk potrafi dostrzegać fakty, matematyk wybitny - analogie między faktami, zaś matematyk genialny - analogie między analogiami." Stefan Banach
"Prawie że nie widziałem matematyka, który byłby zdolny do rozumowania." Platon
They probably haven't met :)))
@ Mathilda S,
ReplyDeleteI don't know what example to look for in files that my computer considers dangerous. Please describe what this example is.
@ A. Jadczyk
Plato's Parmenides - much better. I understand that Zeno's arguments here are different from the paradoxes of motion described by Aristotle: 4.6(133a-134e), 5(134e-137c).
Here is a link to the description and a link to the source in English:
1) https://plato.stanford.edu/archives/spr2020/entries/plato-parmenides/#IntSecZenArg126
2) https://www.wilbourhall.org/pdfs/plato/the_parmenides_of_plato.pdf
I thought syllogisms were invented by Aristotle, but the matter requires investigation.
And there's nothing interesting about this article about 7!=5040? In the abstract on RG there is an error that 7!=5050:
0) https://www.researchgate.net/publication/226417480_Plato%27s_hidden_theorem_on_the_distribution_of_primes
"I don't know what example to look for in files that my computer considers dangerous."
DeleteYou can also get it from here
Correction: references 4.6(133a-134e), 5(134e-137c) concern the proposed examples of mathematical syllogisms in Plato's Parmenides - I did not copy this sentence earlier.
ReplyDelete@ Anonymous,
Nice quotes, those not translated from Polish. The last quote is either far-fetched or indicates that Plato had a different understanding of mathematics, which he nevertheless professed.
The fact that Plato was a philosopher does not exclude that he did something in mathematics - especially since he greatly promoted mathematics. Here you need to look for a good example, not to look for excuses not to look for anything.
The point is that it is claimed that Plato was more mathematical than Aristotle - for example, Krzysztof Meissner, mentioned here, said so somewhere. Others also say so - for example, such a statement results from the first entries of Mathilde S - and there is no point in denying it. I just scientifically verify this theorem with a simple measurement: show me Plato's mathematical example.
@Grzegorz
Delete"The point is that it is claimed that Plato was more mathematical than Aristotle"
---------
This is quite obvious. The psyche - the way of thinking presented - of Plato's is consistent with the modern understanding of mathematics - as so-called, pure mathematics - that is, a set of abstractions/ideas detached from the concrete.
Aristotle in such a comparison is - let's say - a constructivist mathematician -. i.e. an individual closer to concretes. Something like more of an accountant
than a mathematician.
Thus, for idealistic people - delighted by the purity of the
of complete abstraction ( such as the aforementioned M.S. delighted with the category theory ) it is obvious that Plato is more of a mathematician than Aristotle.
In fact, neither Plato nor Pythagoras particularly valued
mathematics - but that's another tale :)))
"In fact, neither Plato nor Pythagoras particularly valued
Deletemathematics"
Experience with this post shows that all the attempts to forcefully reduce the discussion of Plato and Aristotle to mathematics or physics lead to an artificially restrictive and not very productive view. Having this in mind in the next post (probably tomorrow) I will venture to extend the discussion so that the humanistic and esoteric perspective will also appear. Such a perspective is at least as important as that of exact sciences.
@Mathilde S.:
ReplyDelete"the harmonic
mean of c2 is 2"
How come?
"@Mathilde S.:
ReplyDelete"the harmonic
mean of c2 is 2"
How come?".
Sorry, there is probably an error here, as I was transcribing this from the Polish version to the English version and there should be approximations. For c2 this harmonic mean would be 32/15, which is close to 2, but not exactly 2.
@Mathilde S.:
Delete"Sorry, there is probably an error here, as I was transcribing this from the Polish version to the English version and there should be approximations."
Sorry, I doesn't understands your.
"AnonymousJuly 18, 2023 at 8:16 PM".
ReplyDeleteI strongly agree with the previous speaker.
"Having this in mind in the next post (probably tomorrow) I will venture to extend the discussion so that the humanistic and esoteric perspective will also appear. Such a perspective is at least as important as that of exact sciences.".
ReplyDeleteI think the crux of it all here is that in order to make any assumptions about the exact sciences, we need to define the paradigm we are in. Establishing the paradigm, on the other hand, is done on the ground of metaphysics and there is no other way.
This is a large part of the reason why I prefer Kuhn's approach to Popper's. Popper was, in my opinion, short-sighted and forgot about metaphysics and paradigms.
And somewhere at a very deep level, physics and mathematics are indeed art that moves more than painting or music.
@Matilde S.
ReplyDeleteCzytam ten Twój artykuł: "A comparative analysis of ..." i jeśli dobrze rozumiem to przeciwstawiasz sobie dwie średnie harmoniczną i arytmetyczną jakoby mające zupełnie różne odniesienia ontologiczne (metafizyka kontra realność).
To jest dla mnie całkowicie nie do zrozumienia z uwagi na to, że ja nie widzę pojęciowej różnicy między tymi średnimi.
Tak jak średnia harmoniczna jest odwrotnością średniej arytmetycznej odwrotności,
tak i średnia arytmetyczna jest odwrotnością średniej harmonicznej odwrotności.
To czy posługujemy się wielkościami fizycznymi czy raczej ich odwrotnościami jest kwestią przyzwyczajenia i do pewnego stopnia użyteczności.
W szkole np. szczególną uwagę poświęca się pojęciu szybkości a nie pojęciu tempa (odwrotności szybkości). A przecież o ile szybkość przydaje się do obliczenia zadań typu jaką drogę przebył pojazd jeśli jechał z szybkością A przez czas t1 i z szybkością B przez czas t2
to jednak
to właśnie tempo przydaje się do obliczenia zadań typu jaki czas zajęło pojazdowi przebycie drogi s1 + s2 jeśli jechał w tempie C na drodze s1 i w tempie D na drodze s2.
Innym jaskrawym przykładem na brak różnicy pojęciowej między wielkościami i ich odwrotnościami jest "wydajność spalania pojazdu". W Polsce mierzy się ją w jednostce litry/(100km) a w USA w jednostce mile/galon.
TOPIC SUMMARY:
ReplyDeleteInept examples of mathematics in Plato:
a) calculating the age difference in Plato's Parmenides
b) unclear logical relations or syllogisms in Plato's Parmenides (maybe it makes sense, but no one has described it)
c) speculations about the number 5040=7! in Plato's Law
d) unwritten speculations about the arithmetic and harmonic mean
These incompetent examples were born in great pain, the debaters formulated the most bizarre excuses and arguments not to give them. Examples of excuses: the subject is ill-defined, the answer is obvious, Plato had nothing to do with mathematics, Plato is obviously associated with marthematics. In short, a toothache for the supporters of logic - for clarity toothache for me (no mercy for me).
A simple one example of Aristotle's mathematicity:
Aristotle analyzed and developed the concept of infinity in the context of continuity, which may have influenced the development of mathematical analysis. There are four types of infinity in Aristotle:
(i) potential infinity: e.g. enumerating 1, 2, 3, ...
(ii) actual infinity: e.g. sum value 1+2+3+....
(iii) subdivision infinity: e.g. infinite dichotomy of a segment
(iv) the infinity of the ends of the segment: e.g. the ends of the real line
The second infinity pair can be related to the first and vice versa.
RESULT OF THE FIRST ROUND (in mathematics) PLATO vs. ARISTOTLE:
Aristotle knocks out Plato and wins the round and the match.
The supporters of Plato are helpless, but they can think further (whatever they want), but the supporters of Aristotle go home.
I have no more comments - in particular about "Gallus Anonimus".
Thank you for the discussion
Goodbye
"The supporters of Plato are helpless"
DeleteNo, they are not. They happily create very good science.
But it seems to me that it would be productive to have a list of errors of Aristotle as well. You probably know many of them and have them written somewhere. Do you? Would you like to share them with us?
Here is something funny about Aristotle. I am quoting from the internet essay "Aristotle Was Not Wrong about Everything"
Delete"Aristotle’s critics are certainly correct that he could have easily checked to verify whether women really had fewer teeth than men. Nonetheless, I think that much of this criticism is rather unfair. Aristotle was clearly relying on a report he had heard from someone else that he thought was based on observation. Aristotle evidently assumed that the report was correct and did not bother to verify it for himself."
"I have no more comments - in particular about "Gallus Anonimus"."
DeleteArguments regarding Aristotle's analysis of infinity - as an argument for his results in mathematics -. are not very convincing. At that time infinity ( as well as most of science ) was a philosophy - e.g. Zenon's paradoxes.
Infinity appeared in mathematics - really much later - only thanks to G.Cantor.
Consideration of types/types of infinity would fall out as a kind of systematics used by Aristotle to describe many phenomena, e.g. biological phenomena. Is such systematics then mathematics?
I doubt it. But this, of course, everyone can judge according to his own beliefs and needs.
Just out of spite, I will note that Plato - for a change - had a systematics of regular polyhedra - the so-called Platonic solids.
:)))
Wikipedia on this subject tells that:
Delete"The ancient Greeks studied the Platonic solids extensively. Some sources (such as Proclus) credit Pythagoras with their discovery. Other evidence suggests that he may have only been familiar with the tetrahedron, cube, and dodecahedron and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. In any case, Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that no other convex regular polyhedra exist.
The Platonic solids are prominent in the philosophy of Plato, their namesake. Plato wrote about them in the dialogue Timaeus c. 360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron."
Why these solids are called Platonic? It is a good question. But it is a nice name, and I used it too:
Quantum Jumps, EEQT and the Five Platonic Fractals
@Matilde S.
ReplyDeleteI am noticing that the comment: Bjab July 19, 2023 at 11:31 AM is left without answer. It would be beneficial for the readers to see your reply to Bjab's objections....