"Let no one ignorant of geometry enter"

Plato’s Motto

The scho­liast on Aelius Aris­tides 125.14 (Din­dorf, Vol. 3) says the fol­low­ing:

ἐπεγέγραπτο ἔμπροσθεν τῆς διατριβῆς τοῦ Πλάτωνος ὅτι ἀγεωμέτρητος μηδεὶς εἰσίτω· ἀντὶ τοῦ ἄνισος καὶ ἄδικος. ἡ γὰρ γεωμετρία τὴν ἰσότητα καὶ τὴν δικαιοσύνην τηρεῖ.

‘In front of Plato’s school had been inscribed, “Let noone enter un-​​geometried” rather than “unequal” or “unjust,” for geom­e­try main­tains equal­ity and justness.’


At any rate, Pseudo-​​Galen (post 2 A.D.?) quotes the phrase at the begin­ning of ‘On the divi­sions of phi­los­o­phy,’ and makes geom­e­try a pre­lim­i­nary to the­ol­ogy:

ὁ μὲν οὖν Πλάτων εἰς φυσιολογικὸν καὶ θεολογικὸν αὐτὸ διαιρεῖ· τὸ γὰρ μαθηματικὸν οὐκ ἠβούλετο εἶναι μέρος τῆς φιλοσοφίας, ἀλλὰ προγύμνασμά τι ὥσπερ ἡ γραμματικὴ καὶ ἡ ῥητορική· ὅθεν καὶ πρὸ τοῦ ἀκροατηρίου τοῦ οἰκείου ἐπέγραψεν ‘ἀγεωμέτρητος μηδεὶς εἰσίτω’. τοῦτο δὲ ὁ Πλάτων ἐπέγραφεν, ἐπειδὴ εἰς τὰ πολλὰ θεολογεῖ καὶ περὶ θεολογίαν καταγίνεται· συμβάλλεται δὲ εἰς εἴδησιν τῆς θεολογίας τὸ μαθηματικόν, οὗτινός ἐστιν ἡ γεωμετρία.

‘Plato divided it (the­o­ret­i­cal phi­los­o­phy) into phys­i­ol­ogy and the­ol­ogy. In fact, he did not want math­e­mat­ics to be a part of phi­los­o­phy, but a sort of pro­gym­nasma like gram­mar and rhetoric. That’s why, before his pri­vate lecture-​​room, he inscribed “Let no one enter un-​​geometried.” He inscribed this since he dis­coursed on the­ol­ogy in all mat­ters and dwelt on the­ol­ogy, and included math­e­mat­ics, of which geom­e­try is a part, into theology’s forms of knowledge.’ See:Plato’s Motto Written by Dennis McHenry. December 10, 2005

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Polish Society of St. Thomas Aquinas-Plato's Academy

Candidates for philosophy to be properly prepared.

Plato introduction to the philosophy of mathematics has made, highlighting the non-the usual benefits of studying mathematics in the improvement mind. At the front of the AP, as the legend goes, was engraved the inscription: "There is no WStE-pu anyone who does not know geometry. " In the Republic (VII 528 a) Plato classification mathematical sciences conducted on the basis of views Pythagoreans, who shared in the mathematical sciences depending on what questions to give answer: "How much?" - arithmetic and music, "how much?" - geometry and mechanical chanika. Plato arranges in order of mathematical sciences: arithmetic, geometry (distinguished by the geometry of the flat - planimetry and spatial geometry -stereometry), astronomy, music, and considers that these sciences are related to the relation-my formal, uwidocznionymi eg decreasing their abstractness. http://www.ptta.pl/pef/pdf/a/akademiaplaton.pdf

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If the late character of our sources may incite us to doubt the autheticity of this tradition, there remains that, in its spirit, it is in no way out of character, as can be seen by reading or rereading what Plato says about the sciences fit for the formation of philosophers in book VII of the Republic, and especially about geometry at Republic, VII, 526c8-527c11. We should only keep in mind that, for Plato, geometry, as well as all other mathematical sciences, is not an end in itself, but only a prerequisite meant to test and develop the power of abstraction in the student, that is, his ability to go beyond the level of sensible experience which keeps us within the "visible" realm, that of the material world, all the way to the pure intelligible. And geometry, as can be seen through the experiment with the slave boy in the Meno (Meno, 80d1-86d2), can also make us discover the existence of truths (that of a theorem of geometry such as, in the case of the Meno, the one about doubling a square) that may be said to be "transcendant" in that they don't depend upon what we may think about them, but have to be accepted by any reasonable being, which should lead us into wondering whether such transcendant truths might not exist as well in other areas, such as ethics and matters relating to men's ultimate happiness, whether we may be able to "demonstrate" them or not.See: Frequently Asked Questions about Plato by Bernard SUZANNE

Most certainly that given perspective about the reality of geometry in the context  of the abstract,  it is buried deep within ourselves that our creativity leads us that much closer to the truth and points to a depth of our being. Have you not ever been there to know, that by such mapping schematically, any direction lies under the sociological underpinnings of our associations and our dealings with reality?

On any road to self discovery it was apparent to me that by observing levels of awareness that we usually don't take the time to observe, the more I looked, a abstract math of let's say Game Theory, was apparent. When being lead through a mathematical landscape, could we arrive at our everyday dealings in society?

Economically, it had to make sense that such algorithms could be written and many of us as observers of the information world are unaware of the constrains we have applied to our everyday reading of the economic world?

The Synaptic World of Experience and Knowledge

 It is important that people realize that as much as topological seasoning is added to the world by myself,  I see ourselves intrinsically linked to the inductive/deductive process. It is as if the tail of each is linked as a image of a our inner and outer relation with the world continually exchanged. We are the central process in this, as if link the past and the future together, as to the outcome in life. A self eventual recognition of the arche and our place on it as to the decision and acceptance of outcome according to our conclusions?

I think that Fig. 34.1 best expresses my position on this question, where each of three worlds, Platonic-mathematical, physical and mental-has it’s own kind of reality, and where each is (deeply and mysteriously) found in one that precedes it ( the worlds take cyclicly). I like to think that, in a sense the Platonic world may be the most primitive of the three, since mathematics is a kind of necessity, virtually conjuring its very existence through logic alone. Be that as it may, there is a further mystery, or paradox, of the cyclic aspect of these worlds , where each seems to be able to encompass the succeeding one in its entirety, while itself seeming to depend only upon a small part of its predecessor.”(Page 1028-The Road to Reality- Roger Penrose- Borzoi Book, Alfred A. Knoff- 2004)

For me, the visual helps to reinforce some  the understanding that is required of how let's say Sir Roger Penrose may look at the idea of "information transference?" How I may see this in individuals who are interacting with the world. I believe too, that how the universe is formulated into the Cyclical Universe is to direct our attention to the facets of time attached to the ideas of how this is formulated within ourselves as well. This are the same correlations of the past, as well as the future, in our now, in our universe(our neighborhood) as well.

If we can put everything together, we might have a model that reproduces everything we see in our detector."

Plato's problem is the term given by Noam Chomsky to the gap between knowledge and experience. It presents the question of how we account for our knowledge when environmental conditions seem to be an insufficient source of information. It is used in linguistics to refer to the "argument from poverty of the stimulus" (APS). In a more general sense, Plato’s Problem refers to the problem of explaining a "lack of input."

Solving Plato’s Problem involves explaining the gap between what one knows and the apparent lack of substantive input from experience (the environment). Plato's Problem is most clearly illustrated in the Meno dialogue, in which Socrates demonstrates that an uneducated boy nevertheless understands geometric principles.

The understanding here is that all knowledge exists in the universe and that we only have to awaken it within ourselves. This hasn't changed my view on the universal access to information that we can tap into. How is this accomplished.

This view I carry to the world of science and look for correspondences in experimental associations. I believe the answers we are looking for already exist.  It is just a matter of asking the right questions, as well as looking inside as to the truth of what we are looking at,  as a potential in the discourse of our existence as human beings. The role we are playing as components of this reality to better ourselves.

Know Thyself (γνώθι σεαυτόν )

A stained glass window with the contracted version γνωθι σαυτόν.

The saying "Know thyself" may refer by extension to the ideal of understanding human behavior, morals, and thought, because ultimately to understand oneself is to understand other humans as well. However, the ancient Greek philosophers thought that no man can ever comprehend the human spirit and thought thoroughly, so it would have been almost inconceivable to know oneself fully. Therefore, the saying may refer to a less ambitious ideal, such as knowing one's own habits, morals, temperament, ability to control anger, and other aspects of human behavior that we struggle with on a daily basis.

It may also have a mystical interpretation. 'Thyself', is not meant in reference to the egotist, but the ego within self, the I AM consciousness.

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Delphi became the site of a major temple to Phoebus Apollo, as well as the Pythian Games and the famous prehistoric oracle. Even in Roman times, hundreds of votive statues remained, described by Pliny the Younger and seen by Pausanias. Supposedly carved into the temple were three phrases: γνωθι σεαυτόν (gnothi seauton = "know thyself") and μηδέν άγαν (meden agan = "nothing in excess"), and Εγγύα πάρα δ'ατη (eggua para d'atē = "make a pledge and mischief is nigh"),[6] as well as a large letter E.[7] Among other things epsilon signifies the number 5. Plutarch's essay on the meaning of the “E at Delphi" is the only literary source for the inscription. In ancient times, the origin of these phrases was attributed to one or more of the Seven Sages of Greece,[8] though ancient as well as modern scholars have doubted the legitimacy of such ascriptions.[9] According to one pair of scholars, "The actual authorship of the three maxims set up on the Delphian temple may be left uncertain. Most likely they were popular proverbs, which tended later to be attributed to particular sages

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"Let no one destitute of geometry enter my doors." Plato (c. 427 - 347 B.C.E.)

"[Geometry is] . . . persued for the sake of the knowledge of what eternally exists, and not of what comes for a moment into existence, and then perishes, ...[it] must draw the soul towards truth and give the finishing touch to the philosophic spirit."

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III: The "Geometrical Problem" in the Meno.

Further along in the Meno occurs the celebrated case of the Geometrical Example at Meno 87, which in contrast to the previous mathematical illustration, has been twisted, tortured, and intentionally passed over for two centuries. Jebb said (loc.cit.) asven over a century ago:

The hypothesis appears to be rather trivial and to have no mathematical value. . . (which Raven echoes in 1965)

and here follow some barely intelligible geometrical details".

Bluck however, in 1961 devotes an excursus of some sixteen pages to a complete review of views on the problem, which include an array or barely intelligible geometrical details. The passage is made more difficult of interpretation by the fact that Socrates introduces the geometrical example in a very summary manner, which some have felt was an indication or its relative unimportance.

 I believe on the contrary that the almost schematic reference implies that the topic and the example were well known to the Platonic audience, and did not need explanation. Plato knows how to explain in full, and when he refrains we must understand the matter to be common knowledge. The problem as it occurs at Meno 87 a is briefly this:

We will proceed from here on like the geometer who when asked if a given triangle can be inscribed in a given circle, will say:

'I can't say, but let us proceed hypothetically or experimentally, draw out one leg, swing the other two and see if it falls short or exceeds the rim of the circle.'

In making this paraphrase I have added the word "experimentally" for obvious reasons, and I have taken the noun chorion correctly as area (not rectangle or a triangle, as has been said, which means nothing) in a sense very well attested. So apparently with these conditions, the words themselves are not obscure or really unintelligible, although as yet the meaning has not yet come to the surface.

See: Plato: Mathematician or Mystic ?

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You might think the loss of geometry like the loss of, say, Latin would pass virtually unnoticed. This is the thing about geometry: we no more notice it than we notice the curve of the earth. To most people, geometry is a grade school memory of fumbling with protractors and memorizing the Pythagorean theorem. Yet geometry is everywhere. Coxeter sees it in honeycombs, sunflowers, froth and sponges. It's in the molecules of our food (the spearmint molecule is the exact geometric reaction of the caraway molecule), and in the computer-designed curves of a Mercedes-Benz. Its loss would be immeasurable, especially to the cognoscenti at the Budapest conference, who forfeit the summer sun for the somnolent glow of an overhead projector. They credit Coxeter with rescuing an art form as important as poetry or opera. Without Coxeter's geometry as without Mozart's symphonies or Shakespeare's plays our culture, our understanding of the universe,would be incomplete.

See: γνώθι σεαυτόν

Setting Time Aright

Time has no independent existence apart from the order of events by which we measure it. — Albert Einstein

While Event has since past, I hope the lecture itself will remain in public domain. It helps so as to see the context of the discussion provided by this conference with regard to that subject of time.




Video streaming by Ustream

See:Setting Time Aright

In 1952, in his book Relativity, Einstein writes:

Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence

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Setting Time Aright

View more presentations from Sean Carroll

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  If man thinks of the totality as constituted of independent fragments, then that is how his mind will tend to operate, but if he can include everything coherently and harmoniously in an overall whole that is undivided, unbroken, and without a border then his mind will tend to move in a similar way, and from this will flow an orderly action within the whole. (David Bohm, Wholeness and the Implicate Order, 1980)


Lee Smolin:

I suspect this reflects the expectation many people have that time is not fundamental, but rather emerges only at a semiclassical approximation in quantum cosmology. If you believe this then you believe that the fundamental quantities a quantum cosmology should compute are timeless. This in turn reflects a very old and ultimately religious prejudice that deeper truths are timeless. This has been traced by scholars to the theology of Newton and contemporaries who saw space as “the sensorium” of an eternal and all seeing god. Perhaps the BB paradox is telling us it is time to give up the search for timeless probability distributions, and recognize that since Darwin the deep truths about nature cannot be divorced from time.

The alternative is to disbelieve the arguments that time is emergent-which were never very convincing- and instead formulate quantum cosmology in such a way that time is always real. I would suggest that the Boltzman Brain’s paradox is the reducto ad absurdum of the notion that time is emergent and that rather than play with little fixes to it we should try to take seriously the opposite idea: that time is real.

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Bar of Lead Tungstate Source: A Quantum Diaries Survivor-Calorimeters for High Energy Physics experiments - part 1 April 6, 2008

Calorimeters measure the collective behavior of particles traveling along approximately the same path, and are thus naturally suited for the measurement of jets-Dorigo Tommaso

See: 

• FQXi Conference 2011
  
• The Reference Frame: Ten new things modern physics has learned about time

The Truth of Symmetry?

Fidel wrote:

Canadian Tom Harpur wrote that modern day religious scholars began to realize that indigenous people in this hemisphere possessed what are some of the most sophisticated and deeply held spiritual beliefs in the world.

I agree with your perspective about the historical perspective with the introduction of the white people assertions toward the beliefs of their spirituality  and that expressed by the indigenous people of that time. The "new culture"  to squashed that belief.

Over all I see in a  sense what you are saying that what could have befell the indigenous peoples is what could have fell upon the generations that exist now upon the planet by some alien society as an act of evolution and change with those same correlative situations as it did in our past. Many fictions written in this context today.

But again these are the finer mental states of existence of belief structures, polarization and centralization of such beliefs are "matter states" in the conquest of how we may perceive that evolution.

Quote:

LEE SMOLIN- Physicist, Perimeter Institute; Author, The Trouble With Physics

Thinking In Time Versus Thinking Outside Of Time

One very old and pervasive habit of thought is to imagine that the true answer to whatever question we are wondering about lies out there in some eternal domain of "timeless truths." The aim of re-search is then to "discover" the answer or solution in that already existing timeless domain. For example, physicists often speak as if the final theory of everything already exists in a vast timeless Platonic space of mathematical objects. This is thinking outside of time. See:A "scientific concept" may come from philosophy, logic, economics, jurisprudence, or other analytic enterprises, as long as it is a rigorous conceptual tool that may be summed up succinctly (or "in a phrase") but has broad application to understanding the world.

 Lee Smolin does not obviously like the abstractions in the mathematical realm, and prefers to set the pace for scientific realism by denouncing the historical past with regard to foundational approaches by Plato Thales and others who were our forefathers of expression. Like Hawking,  he is seeking to set foot in the realism of today?
So I am struggling with what we can define as "outside time" is no more then the belief of, while in science we are asking to deal with a methodology that is repeatable by expression, so how can we say spirituality is of a kind of substance or can exist amongst all substances and does not exist outside of time?
In dealing with the opinion of Hawking and Smolin I raise the question of Meno,

Quote:

SOCRATES: But if he always possessed this knowledge he would always have known; or if he has acquired the knowledge he could not have acquired it in this life, unless he has been taught geometry; for he may be made to do the same with all geometry and every other branch of knowledge. Now, has any one ever taught him all this? You must know about him, if, as you say, he was born and bred in your house.SEE:Meno by Plato

 

It relates to the question of how the house boy knew what he knew.

Quote:

MENO: And I am certain that no one ever did teach him.

SOCRATES: And yet he has the knowledge?

MENO: The fact, Socrates, is undeniable.

SOCRATES: But if he did not acquire the knowledge in this life, then he must have had and learned it at some other time?

MENO: Clearly he must.

Quote:

Spectrum wrote: Macdougall's mistake was to believe that spirituality could actually weigh something?

Fidel wrote:

I have no idea except to say that some scientists have said that certain phenomenon may not be detectable by our five senses as developed throughout our evolution. Astronomer Lord Rees suggests that we may need to evolve physically and otherwise a lot further in order to fully understand the universe. And I can see that. If we have evolved in a corner of the universe where atomic matter rules, then of course scientists are going to know a lot about physical matter, which is about 4% of everything that there is.

You actually quoted from a post that dissappeared when providing the links for the cultural Books of the Dead. MacDougall reference and materialism. My point was to supply the Macdougall reference to show how spirituality from my perspective is much lighter and really can't be measured in the way you captured the quoted statement I have suggested.

It goes back to the panel I provided about the weighing of the heart against the feather as truth. To me, it is about Gravity and how we are looking at it. Conceptually modelling according to the questions of a theoretical unification of all the forces. Where gravity actually begins and is inclusive. I really have to be careful here of my statements so as to maintain a mainstream correlative thinking that is current and correct. There is so much to remember.

Fidel wrote:

Apparently some pyramidologists believe that if they substitute years for inches, the great pyramid becomes a prophetic calendar of human events culminating in a forked path to our future. A similar dual future is predicted by prophecies according to Hopis of the Americas. Time is a river that flows and branches into forks and even whirlpools according to Einstein.

That's interesting as it related to my own research on that topic. Developing the concept about our "metal imagery" according to some scale was implemented in the construction of the geometry of the pyramid. I tried scaling the substance of thought as a elementary consideration of where and what we grab onto in life so as to show that the harboring of thought in such a domain, reveals how close indeed we court the matter distinctions of the world we live in.

Newton the Alchemist

Newton's Translation of the Emerald Tablet

Quote:

It is true without lying, certain and most true. That which is Below is like that which is Above and that which is Above is like that which is Below to do the miracles of the Only Thing. And as all things have been and arose from One by the mediation of One, so all things have their birth from this One Thing by adaptation. The Sun is its father; the Moon its mother; the Wind hath carried it in its belly; the Earth is its nurse. The father of all perfection in the whole world is here. Its force or power is entire if it be converted into Earth. Separate the Earth from the Fire, the subtle from the gross, sweetly with great industry. It ascends from the Earth to the Heavens and again it descends to the Earth and receives the force of things superior and inferior. By this means you shall have the glory of the whole world and thereby all obscurity shall fly from you. Its force is above all force, for it vanquishes every subtle thing and penetrates every solid thing. So was the world created. From this are and do come admirable adaptations, whereof the process is here in this. Hence am I called Hermes Trismegistus, having the three parts of the philosophy of the whole world. That which I have said of the operation of the Sun is accomplished and ended.

Metal should read Mental, but using the analogy of what Gold men/woman are is as much a part of the allure of chymistry as a novel idea of bettering ourselves is an inclusive thought as well.

Quote:

The Errors & Animadversions of Honest Isaac Newton
by Sheldon Lee Glashow


ABSTRACT:
Isaac Newton was my childhood hero. Along with Albert Einstein, he one of the greatest scientists ever, but Newton was no saint. He used his position to defame his competitors and rarely credited his colleagues.His arguments were sometimes false and contrived, his data were often fudged, and he exaggerated the accuracy of his calculations. Furthermore, his many religious works (mostly unpublished) were nonsensical or mystical, revealing him to be a creationist at heart. My talk offers a sampling of Newton’s many transgressions, social, scientific and religious.

Many did not like this part of Isaac Newton's research but to me even with all the failures of Newtons "mental state" he was trying to better himself, and that is what the Books of the Dead represented to me about trying to understand matter creation in the sense of what we gather together to become who we are. That is a consistent feature of living beings that what gathers around our human feature, gathers around all things? Matter gathers around "a spiritual principal" from inside each of us.

Quote:

The standard model of particle physics is a self-contained picture of fundamental particles and their interactions. Physicists, on a journey from solid matter to quarks and gluons, via atoms and nuclear matter, may have reached the foundation level of fields and particles. But have we reached bedrock, or is there something deeper? Savas Dimopoulos

The Sun then becomes a interesting feature of what "rays of creation" may mean as it gives life to all that falls under it's light that such a light could have existed inside of us as well. That we came from such a place as to exemplify that we are being first of the true signs of spirituality and then become all that falls under these rays of creation.

A refractory status of light itself and Thomas Young's question about spectrum of light and being.
Objects like the pyramid were shadow markers and were part of the history and development of concepts of geometers and angles of Euclid in my views. This is a materialistic explanation while the pyramid itself is a model for understanding the matter creation and sub developmental model of such scaling of human thought. This represented in my views as an attempt to help people of the times to understanding the truth and comparison of what values we hold to heart and our own evolution of being.

Newton Prism Experiment

The pyramidal model of refractions is a display of the light representation of the spectrum of possibilities, as a relational experimental model in my mind of something quite ancient in it's notion,  is applicable in the views of the science today. Something that is forgotten, but as a attempt of all our remembrances whether we like to admit it or not, is a universal understanding of our depth of being and our loss of memory as we are immersed in materiality.

To me this is a aspect of understanding how ideas emerge as if from a world Smolin liked to assign toward "outside of time" and a diversion of the quest to understand aspects of materiality as "not in time." That is my disagreement of him and his thoughts as it relates toward. The "idea of symmetry" and how this is assigned as a relational aspect of the idea of "outside of time." We are of such perfection that such a beauty is simplified in our own existence as spiritual beings that we each contain this within ourselves? This possibility of becoming in the world of materiality.

Life After Death Question

It's obvious that some humour can make light of a dead situation?:)

Stephen Hawking dismisses belief in God in an exclusive interview with the Guardian. Photograph: Solar & Heliospheric Observatory/Discovery Channel

In the interview, Hawking rejected the notion of life beyond death and emphasised the need to fulfil our potential on Earth by making good use of our lives. In answer to a question on how we should live, he said, simply: "We should seek the greatest value of our action." Ian Sample, science correspondent

Seriously though how is it Stephen  can invoke the after life in order to concertize what they are saying about their science. Is this just an affirmation of their scientific position? Later on I raise the question even further with Lee Smolin.

In the quoted paragraph above I agree with the writer when he writes of Stephen Hawking that he," emphasised the need to fulfil our potential on Earth by making good use of our lives ."

I must admit the thought of Meno with regard to Lee Smolin creep into my mind. This in regards to Plato's Problem.

LEE SMOLIN
Physicist, Perimeter Institute; Author, The Trouble With Physics

Thinking In Time Versus Thinking Outside Of Time

One very old and pervasive habit of thought is to imagine that the true answer to whatever question we are wondering about lies out there in some eternal domain of "timeless truths." The aim of re-search is then to "discover" the answer or solution in that already existing timeless domain. For example, physicists often speak as if the final theory of everything already exists in a vast timeless Platonic space of mathematical objects. This is thinking outside of time.

Scientists are thinking in time when we conceive of our task as the invention of genuinely novel ideas to describe newly discovered phenomena, and novel mathematical structures to express them. If we think outside of time, we believe these ideas somehow "existed" before we invented them. If we think in time we see no reason to presume that.

The contrast between thinking in time and thinking outside of time can be seen in many domains of human thought and action. We are thinking outside of time when, faced with a technological or social problem to solve, we assume the possible approaches are already determined by a set of absolute pre-existing categories. We are thinking in time when we understand that progress in technology, society and science happens by the invention of genuinely novel ideas, strategies, and novel forms of social organization. See:A "scientific concept" may come from philosophy, logic, economics, jurisprudence, or other analytic enterprises, as long as it is a rigorous conceptual tool that may be summed up succinctly (or "in a phrase") but has broad application to understanding the world.

See Also: Experiments On Life After Death

Plato's Problem and Meno: How Accurately Portrayed?

SOCRATES: Then he who does not know may still have true notions of that which he does not know?

MENO: He has.

SOCRATES: And at present these notions have just been stirred up in him, as in a dream; but if he were frequently asked the same questions, in different forms, he would know as well as any one at last?

MENO: I dare say.

SOCRATES: Without any one teaching him he will recover his knowledge for himself, if he is only asked questions?

MENO: Yes.

SOCRATES: And this spontaneous recovery of knowledge in him is recollection?

MENO: True.

SOCRATES: And this knowledge which he now has must he not either have acquired or always possessed?

MENO: Yes.

SOCRATES: But if he always possessed this knowledge he would always have known; or if he has acquired the knowledge he could not have acquired it in this life, unless he has been taught geometry; for he may be made to do the same with all geometry and every other branch of knowledge. Now, has any one ever taught him all this? You must know about him, if, as you say, he was born and bred in your house.

MENO: And I am certain that no one ever did teach him.

SOCRATES: And yet he has the knowledge?

MENO: The fact, Socrates, is undeniable.

SOCRATES: But if he did not acquire the knowledge in this life, then he must have had and learned it at some other time?

MENO: Clearly he must.

SEE:Meno by Plato

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LEE SMOLIN
Physicist, Perimeter Institute; Author, The Trouble With Physics

Thinking In Time Versus Thinking Outside Of Time

One very old and pervasive habit of thought is to imagine that the true answer to whatever question we are wondering about lies out there in some eternal domain of "timeless truths." The aim of re-search is then to "discover" the answer or solution in that already existing timeless domain. For example, physicists often speak as if the final theory of everything already exists in a vast timeless Platonic space of mathematical objects. This is thinking outside of time.

Scientists are thinking in time when we conceive of our task as the invention of genuinely novel ideas to describe newly discovered phenomena, and novel mathematical structures to express them. If we think outside of time, we believe these ideas somehow "existed" before we invented them. If we think in time we see no reason to presume that.

The contrast between thinking in time and thinking outside of time can be seen in many domains of human thought and action. We are thinking outside of time when, faced with a technological or social problem to solve, we assume the possible approaches are already determined by a set of absolute pre-existing categories. We are thinking in time when we understand that progress in technology, society and science happens by the invention of genuinely novel ideas, strategies, and novel forms of social organization. See:A "scientific concept" may come from philosophy, logic, economics, jurisprudence, or other analytic enterprises, as long as it is a rigorous conceptual tool that may be summed up succinctly (or "in a phrase") but has broad application to understanding the world.

Plato's Problem

Plato's problem is the term given by Noam Chomsky to the gap between knowledge and experience. It presents the question of how we account for our knowledge when environmental conditions seem to be an insufficient source of information. It is used in linguistics to refer to the "argument from poverty of the stimulus" (APS). In a more general sense, Plato’s Problem refers to the problem of explaining a "lack of input." Solving Plato’s Problem involves explaining the gap between what one knows and the apparent lack of substantive input from experience (the environment). Plato's Problem is most clearly illustrated in the Meno dialogue, in which Socrates demonstrates that an uneducated boy nevertheless understands geometric principles.

Contents 1 Introduction 2 Plato (427 B.C. – 347 B.C.) 2.1 Background 2.2 Early work 2.3 Socrates (470 B.C. - 399 B.C.) 2.3.1 Socratic method 2.4 Meno 2.4.1 Innate knowledge 3 Contemporary parallels 3.1 Linguistics 3.1.1 Universal grammar 3.2 Perception and attention 3.2.1 Visual 3.2.2 Auditory 3.2.3 Subliminal priming 3.3 Long-term memory (LTM) 4 Implications 5 References 6 External links

Introduction

What is knowledge? What is experience? How do they interact? Is there a correlational, causal, or reciprocal relationship between knowledge and experience? These and other related questions have been at the forefront of investigation by problem solvers, scientists, psychologists, and philosophers for centuries. These questions, but particularly the problem of how experience and knowledge interrelate, have broad theoretical and practical implications for such academic disciplines as epistemology, linguistics, and psychology (specifically the subdiscipline of thinking and problem solving). Gaining a more precise understanding of human knowledge, whether defined as innate, experiential, or both, is an important part of effective problem solving.

Plato was the first philosopher who systematically inquired into issues such as those noted above. He wrote many dialogues, such as Euthyphro and the Apology, but it is from the Meno that the modern instantiation of Plato’s Problem is derived. In the Meno, Plato theorizes about the relationship between knowledge and experience and provides an explanation for how it is possible to know something that one has never been explicitly taught. Plato believed that we possess innate ideas that precede any knowledge that we gain through experience.

As formulated by Noam Chomsky,[1] accounting for this gap between knowledge and experience is "Plato’s Problem." The phrase has a specific linguistic context with regard to language acquisition but can also be used more generally.

Plato (427 B.C. – 347 B.C.)

Plato

Background

Plato was born into an aristocratic Athenian family. When Plato was a young man, Athens was defeated in the Peloponnesian War, a tragedy he attributed to the democracy (Russell). Plato was principally opposed to democracy, as he believed "democracy passes into despotism" [2]. Several of the political calamities of the day led Plato to propose an ideal form of government in his most famous work, The Republic, which still has profound influences on modern Western political philosophy.

Early work

Plato’s early philosophical endeavors involved poetry discussing many ideas, such as the differences between knowledge and opinion, particulars and universals, and God and man. These early dialogues do not utilize conventional notions of reason. Rather, they appeal to the emotions, the allegorical, the spiritual, and the mythological interests of an ancient speculative mind.

Controversy surrounds the early dialogues in how they are to be interpreted. Some claim that Plato was truly trying to discover objective reality through these mystical speculations while others maintain that the dialogues are stories to be interpreted only as parables, allegories, and emotional appeals to religious experience. Regardless, Plato would come to formulate a more rigorous and comprehensive philosophy later in his life, one that reverberates in contemporary Western thought to this day.

Some of Plato’s famous works are Phaedo, the Crito, and, as noted earlier, the Meno. Within these works are found a comprehensive philosophy that addresses epistemology, metaphysics, ethics, aesthetics, theology, and logic. As noted, most of the writing is in the form of dialogues and arguments to pursue answers to difficult questions and concepts. Plato’s teacher and mentor, Socrates, always plays a significant and formative role in these dialogues.

Socrates (470 B.C. - 399 B.C.)

Socrates

 

Most of Plato’s philosophical ideas were communicated through his beloved teacher Socrates as a presence in the dialogues. Though Socrates never wrote anything himself, it is evident through Plato’s works that Socrates had an incredible ability to explore the most intense analytical discussions. However, for some there is controversy regarding how much historical fact can be derived from Plato’s Socrates (Russell). Some doubt Socrates ever existed. Others are skeptical as to the accuracy of some of Plato’s dialogues but nonetheless maintain that we can learn a substantial amount of historical information about Socrates from the dialogues. Still others take practically everything Plato wrote about Socrates as veridical history. Regardless, it may be safe to say that Plato never meant to record Socrates verbatim and it may plausibly be concluded that his general ideas were communicated in the dialogues.

Socratic method

As delineated in various writings, the meticulousness, articulation, and sophistication with which Socrates spoke supplies an outstanding problem solving technique – the Socratic Method. The Socratic method may be described as follows: it usually involves others with whom Socrates directly engages (not merely pontificating to an audience), it involves a deep philosophical or ethical question to which an answer was sought, and it usually involves Socrates asking questions either to affirm his understanding of others or to seek their understanding.

If someone disagreed with him, Socrates would execute this process in order to bring about his interlocutor’s reluctant admission of inconsistencies and contradictions. Either Socrates would ask his debators questions about their claims that would lead them to admit their fallacy or Socrates would answer questions by posing questions meant to lead the other to answer their own query.

The Socratic Method

Meno

One such dialogue of Plato’s that utilized the Socratic Method was the Meno. The participants were Socrates, Meno, Anytus, and one of Meno’s slave boys. The dialogue begins with Meno asking Socrates whether virtue can be taught. Socrates responds by stating that he does not know the definition of virtue. Meno replies by stating the characteristics of a virtuous man, to which Socrates responds that the characteristics of a virtuous man may be the by-products of virtuousness but they by no means define virtue. Meno is obliged to agree; to wit, he tries to modify his explanation of virtue. Socrates counters each attempt by pointing to inconsistencies and circular arguments.
Meno seems to commit two fallacies when trying to define virtue. He either defines it using some form of the word itself, or he defines it using other words that call for definitions and explanations themselves. Eventually, Meno is lead to confess his shortcomings as he tries to define the enigmatic term (the Socratic Method is the mechanism that brings about this confession). Socrates claims that a definition of virtue must consist of common terms and concepts that are clearly understood by those in the discussion.
A crucial point in the dialogue is when Socrates tells Meno that there is no such thing as teaching, only recollection of knowledge from past lives, or anamnesis. Socrates claims that he can demonstrate this by showing that one of Meno’s servants, a slave boy, knows geometric principles though he is uneducated. Socrates states that he will teach the boy nothing, only ask him questions to assist the process of recollection. Socrates proceeds to ask the slave boy a series of questions about the size and length of lines and squares, using visual diagrams to aid the boy in understanding the questions. The crucial point to this part of the dialogue is that, though the boy has no training, he knows the correct answers to the questions – he intrinsically knows the Pythagorean proposition.

Innate knowledge

Shortly before the demonstration of Pythagoras’ theorem, the dialogue takes an epistemological turn when the interlocutors begin to discuss the fundamental nature of knowledge. The general question asked is how one can claim to know something when one does not even know what knowledge is. Via the Socratic method, it is shown that the answer to the question posed is innateness - one possesses a priori knowledge.

This is derived from Socrates’ belief that one’s soul existed in past lives and knowledge is transferred from those lives to the current one. "These [ideas] were revealed in a former state of existence, and are recovered by reminiscence (anamnesis) or association from sensible things" [3]. The claim is that one does not need to know what knowledge is before gaining knowledge, but rather one has a wealth of knowledge before ever gaining any experience.

Contemporary parallels

There are contemporary contexts that provide input for the various questions posed here: how to account for the gap between experience and knowledge, what are some of the sources of knowledge, or how much knowledge is possessed prior to experience or without conscious awareness. There are many areas in contemporary linguistics and psychological research that have relevance to these epistemological questions. Linguistic analysis has provided some strong evidence for innate cognitive capacities for language and there are many areas of cognitive psychology that yield hard data from investigations into sources of knowledge. In addition, there are some claims in the Meno that have connections to current research on perception and long-term memory (LTM).

Linguistics

Noam Chomsky

 

Linguistics is the scientific study of language. Chomskian linguistics (an inclusive, though perhaps informal, label for the theories and methodologies of linguistic study spearheaded by Noam Chomsky, meant to encompass his extensive work and influence in the field) includes everything from Chomsky’s earliest work in transformational grammar to more recent work in the Minimalist Program. More exactly, it is the study of the structure of language, or grammar. Chomskian linguistics is defined by a particular theoretical foundation and methodological approach that sets it apart from other linguistic perspectives, such as those described by functional grammar or structuralism (per Leonard Bloomfield) for example. This particular approach to the study of language is also often referred to as Generative linguistics, which is attributed to Chomsky and his early generative grammar work.

Universal grammar

There are several concepts important to the Chomskian (or generativist) approach to linguistics. The most fundamental of these ideas is the theory of universal grammar (UG). Simply put, and as implied by the name, UG refers to those grammatical properties thought to be shared by all (to be universal to all) derivations of human language (anything from Amharic to Zhuang).

Per this conceptualization, UG is innate to all humans – people come "pre-wired" with this universal grammatical structure. A person’s individual grammar (that which is unique to the person) develops from the interaction between the innate universal grammar and input from the environment, or primary linguistic data. This "analytic triplet" (McGilvray, ed., 2005, p. 51), UG + input = grammar, is the functional core of the theory.

Language acquisition

Several questions (or problems) motivate linguistic theorizing and investigation. Two such taken up in Chomskian linguistics are the process of language acquisition in children, and "Plato’s Problem." These subjects are interrelated and viewed as evidence in support of the theory of UG.

One of the simplest ways to approach the concept of universal grammar is to pose a hypothetical question about an aspect of language acquisition in children – why does a child learn the language that it does. As a specific example, how can a child of Asian descent (say, born of Chinese parents) be set down in the middle of Topeka, Kansas and acquire "perfect English?" The answer is that the child does not start with "Chinese," or any other conventionally defined language, in its head. The child does start with general grammatical rules that determine linguistic properties.

Children come equipped with universal grammar, from which any natural human language will develop – without instruction. All that is needed is passive input. If what the child predominantly hears (or sees via sign) as it is maturing through the critical period (in linguistics, that period within which a child must have necessary and sufficient exposure to human language so that language acquisition occurs; without sufficient exposure to primary linguistic data, the UG does not have the necessary input required for the development of an individual grammar; this period is commonly recognized as spanning from birth to adolescence, generally up to 12-years-old, though a shorter or longer critical period is possible on a person-by-person basis) is the English spoken in Topeka, Kansas, then that is what the child will acquire. This is why, regardless of a child’s ethnic/racial background (or any other of a variety of non-relevant factors), the child will know Cockney English, Egyptian Arabic, or isiZulu if the child’s primary linguistic input is Cockney English, Egyptian Arabic, or isiZulu, respectively.

The hypothetical question posed addresses a common misconception about what, exactly, is instantiated in the mind/brain of an individual when it comes to language. It does not address the "logical problem" of language acquisition, i.e., how children transition from ostensibly having no knowledge of language to having full knowledge, in what may be described as a very limited time with apparently limited input.

Plato's problem

To address the issue of apparently limited input, one must turn to what is possibly the most quoted of all arguments in support of universal grammar and its nativist interpretation – Plato’s Problem. The phrase refers to the Socratic dialogue, the Meno; Noam Chomsky is often attributed with coining the term. Plato's Problem particularly refers to a point in the dialogue when Socrates is talking with an uneducated servant and shows, through this interaction, that the servant knows the Pythagorean Theorem though he has never been explicitly taught any geometry. How does the servant know without having ever been taught? Plato’s suggestion is, essentially, that people have innate knowledge.

In the field of linguistics, Plato’s Problem is the problem of finding an explanation for how a child acquires language though the child does not receive explicit instruction and the primary linguistic data (input, or stimuli, from the environment; PLD is necessary for the development of an individual's grammar - language - via input into UG) a child does receive is limited. This limited

Meno

Part of the series on: The Dialogues of Plato

Early dialogues:

Apology – Charmides – Crito

Euthyphro – First Alcibiades

Hippias Major – Hippias Minor

Ion – Laches – Lysis

Transitional & middle dialogues:

Cratylus – Euthydemus – Gorgias

Menexenus – Meno – Phaedo

Protagoras – Symposium

Later middle dialogues:

Republic – Phaedrus

Parmenides – Theaetetus

Late dialogues:

Clitophon – Timaeus – Critias

Sophist – Statesman

Philebus – Laws

Of doubtful authenticity:

Axiochus – Demodocus

Epinomis – Epistles – Eryxias

Halcyon – Hipparchus – Minos

On Justice – On Virtue

Rival Lovers – Second Alcibiades

Sisyphus – Theages

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Meno (Ancient Greek: Μένων) is a Socratic dialogue written by Plato. Written in the Socratic dialectic style, it attempts to determine the definition of virtue, or arete, meaning in this case virtue in general, rather than particular virtues, such as justice or temperance. The goal is a common definition that applies equally to all particular virtues. Socrates moves the discussion past the philosophical confusion, or aporia, created by Meno's paradox (aka the learner's paradox) with the introduction of new Platonic ideas: the theory of knowledge as recollection, anamnesis, and in the final lines a movement towards Platonic idealism.

Contents 1 Characters 2 Virtue 3 Meno's paradox 4 Dialogue with Meno's slave 5 Anytus 6 Concluding dialogue with Meno 7 Meno and Protagoras 8 References 9 External links 10 Bibliography

Characters

Plato's Meno is a Socratic dialogue in which the two main speakers, Socrates and Meno, discuss human virtue: whether or not it can be taught, whether it is shared by all human beings, and whether it is one quality or many. As is typical of a Socratic dialogue, there is more than one theme discussed within Meno. One feature of the dialogue is Socrates' use of one of Meno's slaves to demonstrate his idea of anamnesis, that certain knowledge is innate and "recollected" by the soul through proper inquiry. Another often noted feature of the dialogue is the brief appearance of Anytus, a member of a prominent Athenian family who later participated in the prosecution of Socrates.
Meno is visiting Athens with a large entourage of slaves attending him. Young, good-looking and well-born, Meno is perhaps a sophist from Thessaly, but Plato is not absolutely clear about this. Meno says early on in the dialogue that he has held forth many times on the subject of virtue, and in front of large audiences.

Virtue

The dialogue begins with Meno asking Socrates to tell him if virtue can be taught. Socrates says that he is clueless about what virtue is, and so is everyone else he knows (71b). Meno responds that virtue is different for different people, that what is virtuous for a man is to conduct himself in the city so that he helps his friends, injures his enemies, and takes care all the while that he personally comes to no harm. Virtue is different for a woman, he says. Her domain is the management of the household, and she is supposed to obey her husband. He says that children (male and female) have their own proper virtue, and so do old men -free- or slave, as you like (71e). Socrates says he finds this odd. He suspects that there must be some virtue common to all human beings.

Socrates rejects the idea that human virtue depends on a person's gender or age. He leads Meno towards the idea that virtues are common to all people, that temperance ("sophrosunê"- exercising self control) and justice ("dikê, dikaiosunê"- refraining from harming other people) are virtues even in children and old men (73b). Meno proposes to Socrates that the "capacity to govern men" may be a virtue common to all people. Socrates points out to the slaveholder that "governing well" cannot be a virtue of a slave, because then he would not be a slave (73c,d).

One of the errors that Socrates points out is that Meno lists many particular virtues without defining a common feature inherent to virtues which makes them thus. Socrates remarks that Meno makes many out of one, like somebody who breaks a plate (77a).

Meno proposes that virtue is the desire for good things and the power to get them. Socrates points out that this raises a second problem- many people do not recognize evil (77d,e). The discussion then turns to how to account for the fact that so many people are mistaken about good and evil, and take one for the other. Socrates asks Meno to consider whether good things must be acquired virtuously in order to be really good (78b). Socrates leads onto the question of whether virtue is one thing or many.

No satisfactory definition of virtue emerges in the Meno. Socrates' comments however show that he considers a successful definition to be unitary, rather than a list of varieties of virtue, that it must contain all and only those terms which are genuine instances of virtue, and must not be circular.^[1]

Meno eventually throws up his hands at the problem, confessing that he is no longer so sure of what virtue is. He seems agitated and compares Socrates to a torpedo fish that can numb, and admits that he is "quite perplexed". He says that Socrates has made him numb in his mind and tongue (80a,b). Socrates argues that the reason for this comparison is that Meno, a "handsome" man, is inviting counter-comparisons because of his own vanity. Socrates tells Meno that he only resembles a stingray if it numbs itself in making others numb (80c).

Meno's paradox

Socrates brings Meno to aporia (puzzlement) on the question of what virtue is. Meno responds by accusing Socrates of being like a 'torpedo fish' (meaning some species of electric ray), which stuns its victims. Meno then proffers a paradox, "And how will you inquire into a thing when you are wholly ignorant of what it is? Even if you happen to bump right into it, how will you know it is the thing you didn't know?" (80d1-4). This paradox has troubled philosophers for generations.

Meno asks Socrates how a person can look for something when he has no idea what it is. How can he know when he has arrived at the truth when he does not already know what the truth is?(80d) Socrates avoids this sophistical paradox by pointing out that, by using this logic, man could neither search for what he does know, because he would already know it, nor for what he does not know, because he would not know for what he was looking. (This is not to say that Socrates or Plato deny the strength of this paradox. Socrates does not take the position that knowledge can be sought; as is explained, his theory of knowledge is that it is never acquired, only ever recollected.)

Plato subsequently discusses his own theory of knowledge through Socrates, that it is "recollection" from the past lives of the immortal soul.(81d)

Dialogue with Meno's slave

The blue square is twice the area of the original yellow square

Socrates does not answer the paradox. Instead he counters with a mythos (poetic story): the theory of recollection. It tells of our immortal souls mixing with the objects of knowledge in a great world-spirit. Since we have contact with real things in this stage of existence prior to birth, we have only to 'recollect' them when alive. Such recollection requires Socratic questioning, which according to Plato is not 'teaching.' Socrates thus demonstrates his method of questioning and recollection by interrogating a slave who is ignorant of geometry. The subsequent discussion shows the slave capable of learning a complex geometry problem, because "he already has the questions in his soul." In this way, Socrates shows Meno that learning is possible through recollection, and that the paradox is false. Meno's aporia shows that he is not capable of learning, but the examination of the slave shows that he can learn. In this way, Socrates answers Meno's paradox with one of his own. Not understanding, Meno attempts to avoid Socrates' ironic criticism by returning to his original question: "what is virtue?"
Socrates begins one of the most influential dialogues of Western philosophy regarding the argument for innate knowledge. By drawing geometric figures in the ground Socrates demonstrates that the slave is initially unaware of how to find twice the area of a square.
Socrates then said that before he got hold of him the slave (who has been picked at random from Meno's entourage) has spoken "well and fluently" on the subject of a square double the size of a given square (84c). Socrates comments that this "numbing" he caused in the slave did him no harm (84b).

Socrates then draws a second square figure on the diagonal so that the slave can see that by adding vertical and horizontal lines touching the corners of the square, the double of its area is created. He gets the slave to agree that this is twice the size of the original square and says that he has "spontaneously recovered" knowledge he knew from a past life (85d) without having been taught. Socrates is satisfied that new beliefs were "newly aroused" in the slave.
After witnessing the example with the slave boy, Meno tells Socrates that he thinks that Socrates is correct in his theory of recollection, to which Socrates replies, “I think I am. I shouldn’t like to take my oath on the whole story, but one thing I am ready to fight for as long as I can, in word and act—that is, that we shall be better, braver, and more active men if we believe it right to look for what we don’t know...” (86b). It has been argued variously that this implies Socrates is skeptical regarding knowledge or that he is a pragmatist. It also prepares us for the subsequent discussion of knowledge by hypothesis.

Anytus

When Anytus appears, Socrates praises him as the son of Anthemion, who earned his fortune with intelligence and hard work. He says that Anthemion had his son well-educated, and Anytus, the beneficiary of a well-meaning father, must both be virtuous and know what it is. Anytus comments on Sophists, and saying that he neither knows any, nor cares to know any. Socrates then questions why it is that men do not always produce sons of the same virtue as themselves. He alludes to other notable male figures, such as Themistocles, Aristides, Pericles and Thucydides, and casts doubt on whether these men produced sons as capable of virtue as themselves. Anytus becomes offended and accuses Socrates of slander, warning him to be careful expressing such opinions. (The historical Anytus was one of Socrates' accusers in his trial.)

After speaking to Anytus, Socrates suggests that Anytus does not realize what slander is, and continues his dialogue with Meno as to the definition of Virtue.

Concluding dialogue with Meno

After the discussion with Anytus, Socrates and Meno return to the subject of whether Virtue can be taught. Socrates points out the similarities and differences between "true beliefs" and "knowledge". He claims that while "true beliefs" may be as useful to us as knowledge, they often fail to "stay in their place" and must be "tethered" by what he calls aitias logismos (the calculation of reason, or reasoned explanation), immediately adding that this is anamnesis, or recollection.^[2] Whether or not Plato intends that the tethering of true beliefs with reasoned explanations must always involve anamnesis is explored in later interpretations of the text.^[3][4] Socrates' distinction between "true beliefs" and "knowledge" forms the basis of the philosophical definition of knowledge as "justified true belief". Myles Burnyeat and others, however, have argued that the phrase aitias logismos refers to a practical working out of a solution, rather than a justification.^[5]

"To sum up our enquiry," Socrates concludes, "the result seems to be, if we are at all right in our view, that virtue is neither natural nor acquired, but an instinct given by God to the virtuous." In most modern readings these closing remarks are "evidently ironic",^[6] but Socrates' invocation of the gods may be sincere, albeit "highly tentative".^[7]

Meno and Protagoras

Meno's theme is also being dealt in the dialogue of Protagoras, where, Plato finally puts Socrates to conclude with the opposite conclusion 'That virtue can be taught.' And, whereas in the Protagoras knowledge is uncompromisingly this-worldly, in the Meno the theory of recollection points to a link between knowledge and eternal truths.^[8]

References

1. ^ Jane Mary Day. Plato's Meno in Focus. Routledge, 1994, p. 19. ISBN 0415002974.
2. ^ Gregory Vlastos, Studies in Greek Philosophy: Socrates, Plato, and their tradition, Volume 2, Princeton University Press, 1996, p155. ISBN 069101938X.
3. ^ Gail Fine, Inquiry in the Meno, in Richard Kraut, The Cambridge Companion to Plato, Cambridge University Press, 1992, p221. ISBN 0521436109
4. ^ Charles Kahn, Plato on Recollection, in Hugh H. Benson, A Companion to Plato, Volume 37, Wiley-Blackwell, 2006, p122. ISBN 1405115211.
5. ^ Gail Fine, Knowledge and True Belief in the Meno, in David Sedley, Oxford Studies in Ancient Philosophy: Volume XXVII: Winter 2004, Oxford University Press, 2004, pp61-62. ISBN 0199277125
6. ^ Robin Waterfield in Plato, Oxford World Classics: Meno and Other Dialogues, Oxford University Press, 2005, pxliv. ISBN 0192804251
7. ^ Dominic Scott, Plato's Meno, Cambridge University Press, 2006, p193. ISBN 0521640334
8. ^ Jane Mary Day. Plato's Meno in Focus. Routledge, 1994, p. 10. ISBN 0415002974

External links

Wikisource has original text related to this article: Meno

• Text of "Meno" on Project Gutenberg
• Meno on the Perseus Project translated by W.R.M. Lamb (1967) ISBN 0674991834, 0674991842
• Approaching Plato: A Guide to the Early and Middle Dialogues
• Guides to the Socratic Dialogues: Plato's Meno A beginner's guide to the Meno

Bibliography

• Klein, Jacob. A Commentary on Plato's Meno. Chapel Hill: University of North Carolina Press, 1965.