Studies, such as by Godel, on incompleteness imply all conceptual systems, including logic and mathematics, and, by extension, the sciences, are incomplete.
I see knowledge as a state of awareness, a partial understanding in the mechanisms behind nature. This understanding must not only well capture the relation between several influential factors, but also acknowledge the limitations of reducing a dynamical and chaotic system to a mathematical model that is independent from many known and unknown factors equally present in nature. In addition, I think that reality merely exists as a common, shared perception (e.g. we agree to a specific set of theories that we find relevant in our experience as humans) and is very different from something that you would define as a unified theory that explains everything (whatever that may be). However, if you choose to use a different definition of reality, these interpretations may clearly change.
In general, I would agree with the statement that knowledge is the understanding of existing theories, and does not directly reflect understanding of reality. Although there must be some relation between both, it remains unclear to me to what extent knowledge truly captures reality (even when limitations are clear).
I see knowledge as a state of awareness, a partial understanding in the mechanisms behind nature. This understanding must not only well capture the relation between several influential factors, but also acknowledge the limitations of reducing a dynamical and chaotic system to a mathematical model that is independent from many known and unknown factors equally present in nature. In addition, I think that reality merely exists as a common, shared perception (e.g. we agree to a specific set of theories that we find relevant in our experience as humans) and is very different from something that you would define as a unified theory that explains everything (whatever that may be). However, if you choose to use a different definition of reality, these interpretations may clearly change.
In general, I would agree with the statement that knowledge is the understanding of existing theories, and does not directly reflect understanding of reality. Although there must be some relation between both, it remains unclear to me to what extent knowledge truly captures reality (even when limitations are clear).
Reality, as Godel found out, is infinite. Any knowledge of it, however, must be finite. Theories simply are generalizations deduced from sensory impressions of the universe. To survive, the microcosm of the brain never needs to "truly capture reality." It only needs just enough knowledge to make the next move that allows it to exist for a few more microseconds.
http://scientificphilosophy.com/
@ Prakash: Facts are not isolated, they have to be embedded within some conceptual model. Otherwise we would forget the facts we "know". Memory is associative.
It appears that most of the comments are in agreement.
Now, what we have to do with this understanding is to constantly remind ourselves to be less dogmatic and be open to other points of view. In fact, we need to respect one another's point of view (no matter how odious it may appear to us), assuming that each party is acting in good faith and is simply expressing the imperatives of their assumptions. We may try to convince them of our (perhaps transient) point of view; or listen and modify our own point of view. We also need to ask about the other's basic assumptions and goals, or uncover with them the many hidden assumptions they may be making.
All sincere people have valid points of view or models - from their perspective. So, maybe we should try to find the common ground that we can agree on, such as the importance of family, and then start building from there (while not denigrating the other person or his/her belief system).
It is interesting to me that the discussion of knowledge above has referred exclusively with theory or the explanations of relationships or phenomena. That certainly is part of knowledge and the part we apparently are most fascinated with. However, it is not all of knowledge, and not even the part that occupies almost the entire content of lecture and lab courses in science. What I refer to are the laws and rules of nature that we have discovered such as the law of gravity, the conservation of mass in chemical reactions, the laws of thermodynamics, Ohm's and Kirchoff's laws, etc. These are expressed as mathematical formulae among quantities we have defined and can often measure. We learn how to apply these laws to design devices and bridges, guide satellites around the solar system, and devise tests of proposed theories. This certainly is a kind of knowledge that is applied without an explicit need to have an explanation of why they are what they are. They are assumed to truly and accurately describe the relevant phenomena within the boundaries over which they have been tested. In other words, there is the "what" and the "why."
As to the "why," all our explanations of how nature works are, as others have agreed, are models that help us picture the process and make sense of it. They are partial and subject to revision and replacement. If they are useful models, they will suggest experiments which will support of refute them. If the result is positive, we way that the theory has been "verified" but this is misleading as it has not been proven to be true. A better word would be "substantiated." Thories can become more robust or useful as they are tested, extended, and coincide with other theories. Theories that do not include a model that enables further tests or predictions of phenomena are not scientific. If scientists more generally understood that a theory that "works" is neither true nor unique, we would be more open to breakthrough concepts and new discovery.
These comments relate specifically to knowledge of how nature works. There are other areas of knowledge including mathematics, language, logic, etc.
It is true reality is very comlex and whatever we coprehend and uderstan becomes knowledge. If some of observed facts and processes form a system or pattern, it becomes a model. But, it is simplified version of reality as uncertainty cannot be ruled out and complexity cannot be denied. However, these simple models of reality/nature are not useless till these help us in controlling and perdicting nature or outcome of an act or inventing something useful on the basis of these models. This is the way human civilisation has progressed so far. These models are also helpful to build upon them and go deeper. However, absolute knowledge is impossible.
But Mohammad, What I am saying is that there are rules we use in experiment design and product engineering that we count on as being resliably true. Newton's laws of motion and Ohms law relating voltage, current and resistance are such. They are not models. They do not explain why the phenomenon is what it is, they just formalize the observed relationships. It is the explanation of the phenomena such as what gravity is and how charge is conducted that are the models. The laws will continue to work even as the theories of the processes they formulate are revised or replaced. Coming up with a new theory of conductance will not result in Ohm's law suddenly not working. We still use Newton's laws of motion when the velocities are not relativistic. Our culture is obsessed with the theory to the point where there is little attention paid to the laws which we count on working every day. Science and engineering courses are excersizes in the application of the laws, not the development of new theories. Yet, when we think of scientific discovery, it is the new hypothesis that gets the attention.
Our failure to make the distinction between these confirmed relationships and the theories of why they should be so is a major contributing factor to the conflicts among scientists and between scientists and the public. We are defending the theories as true which they aren't and ignoring the confirmed relationships which are widely used products of the scientific process.
In Section VI of the Discours de métaphysique 1686: Leibniz considers "experimental data" obtained by scattering spots of ink on a piece of paper by shaking a quill pen. Consider the finite set of data points thus obtained, and let us ask what it means to say that they obey a law of nature. Well, says Leibniz, that cannot just mean that there is a mathematical equation passing through that set of points, because there is always such an equation! The set of points obey a law only if there is a simple equation passing through them, not if the equation is "fort composée" = very complex, because then there is always an equation.
In other words,
understanding is compression and comprehension is compression.
The determination of what it takes to become a useful law of nature is not the degree of complexity of the equation relating the observed quantities and properties, but its repeated appearance in nature. Liebniz was looking at just one data set and one which was not reproducible at that. I am leaning toward the use of the term "confirmed relationship" rather than "law" just to avoid such confusion.
“All meaning lies in higher levels of reality that are not reducible to the laws by which the ultimate particulars of the universe are controlled…What is most tangible has the least meaning and it is perverse then to identify the tangible with the real.
For to regard a meaningless substratum as the ultimate reality of all things must lead to the conclusion that all things are meaningless. And can avoid this conclusion only if we acknowledge instead that the deepest reality is possessed by higher things that are least tangible.”
Michael Polanyi
The striving for ultimate reality or truth is futile as there will always be a multitude of levels that are beyond our current understanding and we are limited by perception and expression. This dismal conclusion, however, denies that somehow, we have studied nature and have created knowledge about the part of nature we can observe that drives both the production of devices, materials, and machines, but also aids in the further pursuit of scientific knowledge. Is this knowledge ultimate truth or ultimate reality? Of course not. But it certainly does have value and the application of this knowledge has totally transformed (for better or worse) our lives on this planet. On a practical level, I subscribe to a division of that knowledge similar to that of Matson (Grand theories and Everyday Beliefs). He sees everyday knowledge such as the rules of arithmatic and the laws of thermodynamics as low beliefs and which i call rules or laws or confirmed relationships. The explanations for these relationships he calles high beliefs and which are conjectural, and metaphorical views on why things are the way they are. It is the high beliefs that are subject to revision by science or society.
Thomas Brody ( book: The Philosophy Behind Physics) makes an analogy between the relation of a map and the terrain it depicts on the one hand and the relation of the predictions of a physical theory and the reality of the space-time events it predicts, on the other. He uses this analogy to show the limits of our knowledge with a beautiful discussion of a map of the London subway. This map has little if any resemblance to the actual steel rails of the physical system, and yet is very useful to those traveling on that subway. The map describes some aspect of reality but is not reality.
Excellent discussion all of you. Louis, I particulary like the analogy that you quote about the London subway. It very clearly illustrates what we all have to keep in mind, especially if we want fully understand why various critical systems do not work well. They are often based on models that are assumed to be applicable, but no longer are because conditions have changed. Such considerations are what lead to breakthroughs or paradigm changes, which is what society needs to help solve the many problems facing the world today.
Knowledge, the understanding is our comprehension of the world through perceptual experience or enhanced perception/observation that is organised in our mind. This organisation results into conceptualisation of things, ultimately into models of nature of things/phenomena and as how are things are related and works. If the model holds true through observations and experiments, it becomes theory. Theories come mostly in two forms: phenological and explanatory. On the basis of recurrence of phenomena, phenological theories make generalisation. On the other hand, explanatory theories attempt to explain phenomena and processes. In this process of explanation an already tentative explanation or hypothesis is tested by making observations, measurements and experiments. Naturally, observations, measurements and experiments are made according to hypothesis. That is, whatever so-called scientific method is used it always uses only relevant or theory-laden or theory-dictated information. In this sense science is based on a reductionist view of reality. It is not possible by human mind to take into consideration all the fact into consideration as we are grappling with reality in dark. The same applies to standard model of particle physics that seizes only part of the reality. Being a geographer, I fully appreciate Louis’ analogy of the London subway.
However, in present time wherein everything is measured by the yardstick of wealth even success of science and people, who bothers about higher level of reality.
However, it should be clear in our mind as stated in my holy scripture, the Holy Qur’an that humans are created with limited knowledge. Therefore, there is no way to have complete or absolute knowledge. One has only to follow the path as shown by a long series of prophets for a peaceful, equitable and just world.
But, a limited knoledge of the reality does not entail one to leave the pursuit of knowledge as it is helpful in solving many of problems faced by humanity, if not possessed to make money or denied to others for their betterment.
Science will always just be the tip of the iceberg of human knowledge. The human mind, in the terminology of computer scientists, is "massively parallel" meaning it can think many different things at the same time, even things that science would regard as contradictory with one another. These massively parallel thoughts are then blended with prior models (parables or stories we have lived or heard) and with emotions to create real knowledge. The most important knowledge always has these ingredients ... and even science is driven by the instinctive yearning to reach the be core truths that are compatible with the inner parables and landscape. Another point is that a significant part of the thinking apparatus is contained external to the individual, in the group mind. ---------Regards, James
Please excuse this long comment.
I have ridden the London subway and used the subway map. I have also read Brody. The London subway map concept has been copied in many cities around the world. It assumes that the physical relationship of the stops to each other and the distance between them are things the user either already knows or doesn't need. When negotiating my way in an unfamiliar city, I need to have a physical map with the stops marked on it to actually know which routes and stops are closest to my destination. I guess the main point of the subway map analogy is that we only observe a portion of the total reality that exists. But in nature, as opposed to the subway system, we can never have a complete map of the location of every rail, spike, and electrical connection. This supports the final conclusion that there is no such thing as absolute truth and that our knowledge of nature will always be incomplete. I totally agree.
However, I believe it is of great value, in the discussions of scientific discovery and in the teaching of science, to distinguish between what you all are calling the models and I am calling the explanations on the one hand, and the mathematical relationships among measurable and defined quantities of natural phenomena on the other. It is true that neither of these aspects of knowledge can claim to be ultimate truth, but the bases of their shortfall in this regard are very different.
In the case of confirmed relationships (CR) like Fg = Gm1m2/r^2 for the force of gravity, they are reliably applied over a wide range of circumstances. We do not know if the force of gravity equation works outside the solar system or in intergalactic space for sure. However we know that CR’s are valid over the range of conditions for which they have been tested and under well-defined conditions for which they is appropriately applied. These CR’s are at the heart of every science class, lab and problem set. They are the basis of all applied science upon which a huge fraction of our society now depends. There is nothing hypothetical about them. Furthermore, they do not depend on any particular explanation of the related phenomena for their useful application. In many cases, a useful CR has been developed well in advance of any plausible explanation.
On the other hand, explanations and models are our ways of envisioning the process so that it makes sense to our human brains. It is this making sense that enables the devising of tests for the theory, extensions of the theory, and the interrelationships among theories. However, each theory is a picture or metaphor of a particular small piece of nature. It is fragmentary and subject to revision and replacement. For some strange reason, it is the explanations that are most intriguing to us as humans and so it is the development of a new theory that gets the most attention. But it is the unsung CR’s who are doing all the work.
Have we discovered the Higg’s boson and confirmed its existence? It has not been seen or characterized. An accepted theory has proposed that if it existed, its effects would be such and so. Finally, we have observed a reaction for which the effects are such and so. Therefore the Higgs exists? The existence of a theory that explains a set of data supports the hope that the data are consistent and truly reflect the phenomena observed. However, that does not preclude other equally valid explanations for the same set of data. The fitting of a data set to a predicted equation is not absolute proof of the theory that made that prediction. In fact, from Poincare and others, we should expect multiple explanations for the same set of data. Perhaps an alternative theory would postulate a different cause of the “such and so” effect. Without more data, we will not know.
@ Chris: You bring up some interesting points. Basically, we see phenomena and we ask questions like how, why, what, under what conditions, when, under what assumptions, ... ? These questions then provoke our hypothesizing a model. The model is then CAREFULLY put to an experimental test to see if the experimental results were predicted or were consistent with the hypothesized model. If so, then we have some evidence in support, and it is left to other experimenters to confirm our findings. If not, then we have to ask if we performed the experiment correctly, under the right conditions, etc. If we can find no flaw in our experiment design or execution, then we may claim that the experiment tends to deny the hypothesized model.
A model that is confirmed by experimentation, as long as all the right conditions are met, is accepted as a theory or a law. However, we must keep in mind the circumstances and conditions subsumed in the theory or law. When you violate the conditions or circumstances, you are on your own - most likely the theory or law will not predict an outcome precisely.
For example, consider static friction. If, say a block is sitting on a surface, and you push the block, a certain amount of force is needed to make the block move. HOWEVER, you will likely get a different answer for the amount of force needed to move the block, depending on how long the block has been in contact with the surface - one minute or one century. In the case of one century, maybe molecular diffusion has taken place between the block and the surface - the degree of diffusion depending on the types of materials - making it harder to move the block.
@ James: Your comment about our massively parallel brains is well taken. It is hard for us to model our parallel thinking, so we need to break it down into a hierarchical assembly of linear thought processes - a time consuming and challenging task. Artificial neural networks offer another approach model our parallel thinking, but the A.I. investigator has only modeled the architecture, and from this he/she can observe the behavior of the system. However the investigator is limited to viewing the network at a meta-level.
But your main point, I think, is that we can only focus attention on one of many of the parallel processing paths. The path that is most useful to us at that time. AND we forget about the other paths. As a result, we build our models of reality using the single chosen path, BUT we leave the other dimensions of our perceptions and mental processing of reality behind, and often, forgotten.
Congrats,all of you.Mature and meaningful.Deserves to be a model for other threads.Please keep contributing.
@ Larry: Very interesting points you bring up. Two ideas occur to me: (1) In Hofstadter's Book, "Godel, Escher, Bach," he demonstrates that symbols do not have any meaning in and of themselves; but rather the observer, the interpreter brings meaning to the symbols. By extension, I would suggest that works of art, nature's creations, likewise do not have meaning in and of themselves, but the observer, the interpreter brings meaning - if merely the name of the thing observed. On the other hand, (2) when a mystic in an "I-Thou" perceptive state, an innocent child, or some artists observe things, including nature's creations, they are not merely observing, and they are definitely not interpreting, - they are becoming one with the thing.
I would say that neither state of mind is superior to the other. It is simply a matter of how the individual wants to dedicate his limited resources (the brain) at a given moment. It is best, of course if the individual can, of his/her own volition, switch the way he/she views an object. The undenial power of analytic thinking can be brought to bear AND the greater awareness of, normally, hidden properties or patterns exists in a mystical view which can provide the insight for a needed paradigm changes.
@Larry: Based on my personal experience, the second case: "symbolic logic ceases to be relevant" is an accurate description of what happens when in an altered state of mind.
In case you have had no direct experience with a mystical state of mind and its after-effects, there are a few things to to clarify: 1) There is the intense experience of union with the universal spirit - a blissful dying-like experience; 2) There are the after-effects which are not as intense, but which can be characterized as being in a passive, I-Thou relationship with the world - for example, you feel everyone's joy, you become one with the chair you sit in, etc. There is almost no separation between you and what you sense.
However, in such a state, when it comes to symbolic logic, you find the shape, texture, hue of each symbol of interest, rather than what they represent. (This way of looking at the world is reputed to be the reason that Pablo Picasso did poorly in elementary school.)
For a balanced way of life, your prefrontal cortex needs to be able to decide when it is appropriate (with respect to the priorities and context it has established for a given moment in time) to suppress the logical part of your thinking and allow the intuitive state of mind to occupy your full attention; or vice versa.
It is a choice you make. You cannot be in both states of mind simultaneously. But you are most complete and effective, if you can switch from one state of mind to the other, as appropriate to the situation at hand.
Dear Antonio,
Possibly my work may be of interest to you:
TIMAEIC PLATO’S TRIANGLES AND RELATED GEOMETRIES by PANAGIOTIS STEFANIDES
I would like to share with this work, giving a different approach to the interpretation of Plato’s Most Beautiful Triangle:
In section 53, of PLATO'S "TIMAEUS", PLATO speaks about the triangular shapes of the Four Elemental Bodies, of their kinds and their combinations:
These Bodies are the Fire (Tetrahedron) the Earth (Cube), the Water (Icosahedron), and the Air (Octahedron). These are bodies and have depth. The depth necessarily, contains the flatsurface and the perpendicular to this surface is a side of a triangle and all the triangles are generated by two kinds of orthogonal triangles: the "ISOSCELES" Orthogonal and the "SCALΕΝΕ" Orthogonal. From the two kinds of triangles the "Isosceles" Orthogonal has one nature. (i.e. one rectangular angle and two acute angles of 45 degrees), whereas the "scalene" has infinite (i.e. it has one rectangular angle and two acute angles of variable values having, these two acute angles, the sum of 90 degrees). From these infinite natures we choose one triangle "THE MOST BEAUTIFUL".
Thus, from the many triangles, we accept that there is one of them "THE MOST BEAUTIFUL.
Let us choose then, two triangles, which are the basis of constructing the Fire and the other Bodies : "Το μεν ισοσκελές, το δε τριπλήν κατά δύναμιν έχον της ελάττονος την μείζω πλευράν αεί."
PANAGIOTIS STEFANIDES INTERPRETATION:
One of these two is the "ISOSCELES" orthogonal triangle, the other is the "SCALENE" orthogonal triangle, its hypotenuse having a value equal to the "CUBE" of the value of its horizontal smaller side and having its vertical bigger side the value of the "SQUARE" of its smaller horizontal side. The value of the smaller horizontal side is equal to the square root of the Golden Number, the ratio of the sides is equal, again, to the Square Root of the Golden Number (geometrical ratio) and the Tangent of the angle between the hypotenuse and the smaller horizontal side is also equal to the Square Root of the Golden Number (Θ =51 49-38-15-9-17-19-54-37-26-24-0 degrees). The product of the smaller horizontal side and that of the hypotenuse is equal to the "SQUARE" of the bigger vertical side, and the following equation holds:
T^4-T^2-1=0 , T = SQRT [(SQRT.(5) + 1)/2].
In connection with other theorems, it could prove useful to researchers.
Ref: http://www.stefanides.gr/Html/GOLDEN_ROOT_SYMMETRIES.htm
http://www.stefanides.gr/Html/gmr.htm
© Copyright 1986 - 2011 P. C. Stefanides.
N.B. Kepler [ Magirus] Triangle is a similar one but not the same:
http://en.wikipedia.org/wiki/Kepler_triangle
Further to my writing on the "Timaeic Plato’s Triangles", I would like to give the rationale behind my interpretation of the special scalene triangle differing from that of the current interpretation of the 30-60 deg one.
1) Text in Ti. 55, A -54 D,E talks about the structure of the Solid of the First Kind , built by 4 Equilateral triangles each being built by 6 scalene ones [ 30-60 deg] .For this triangle it is clearly stated that its HYPOTENUSE is double its SMALLER side.
2)in Ti.54B the text gives again? the definition of the 30-60 deg [according to the current translation] by a very obscure description ".... TRIPLIN KATA DYNAMIN ECHON TIS ELATTONOS TIN MEIZO PLEYRAN AEI.." WHY?
Here the current interpretation accounts for a "SQUARE of the BIGGER SIDE" of the orthogonal, whereas my interpretation the text states: the "SIDE" and NOT " THE SQUARE OF THE SIDE".
3) The text talks about the :BIGGER SIDE" and the "SMALLER SIDE" leaving to me to make comparison with the third side of the scalene orthogonal as the "MEAN" side, between a "BIGGER" side and a "SMALLER" side.
4) Making use of the Philosophical Ratios : FIRE/AIR = AIR/WATER=WATER/EARTH interpreted as T^3 / T^2 =T^2 /T^1 =T
the obtained triangle fits to the equation: T^4 - T^2 -1 =0, T = SQRT [(SQRT.(5) + 1)/2].
ALL TRIANGLES DERIVE THEIR ORIGIN FROM TWO TRIANGLES
[ Pl.Ti.53D] D].
http://www.stefanides.gr/html/All_triangles_d.rive.htm
....all triangles derive their origin from two triangles, each having one angle right and the others acute....These we lay down as the principles of fire and all the other bodies...Pl.Ti. 53 D.
One of these two triangles is of course the 45/45 deg [tan 45 equals 1] orthogonal triangle and my interpretation of the other scalene orthogonal is that whose angle between hypotenuse and the horizontal smaller has as tangent the sqrt[Φ], contrary to the currently accepted interpretation of the 30/60 deg scalene orthogonal one.
We consider the following:
sqrt[Φ]= sqrt[[sqrt(5) + 1 ]/2] = 1.27201965…= T , with corresponding angle θ, so that:
θ = Arctan(T) = 51.82729238… deg.
An implementation of the relationships involving the two triangles [angles tan(45) deg. and tan(θ)], is:
Sqrt[ [tan(45)+sqrt[5*tan(45)]*cos(60)] = tan(51.82729238…)=1.27201965..=T =sqrt[Φ], or
[1+sqrt(5)] / 2] = Φ,
where, Φ = 1.618033989… , with corresponding angle φ, so that:
φ = Arctan(Φ)= Arctan(T^2) = 58.28252559…deg.
θ = Arctan(T)
θ = 51.82729238… deg., and
Tan [θ] = T ,
Tan [φ] = T^2= 1.618033989..=Φ
[Tan [θ]]^2 = Tan [φ], or
Tan [θ ]= sqrt[Tan [φ]]
θ = Arctan[sqrt[Tan[φ]]]
Angle 60 deg. [the constituent equilateral triangle angle[Pl.Ti.54 E],
formed by 6 scalene orthogonal triangles of the 30/60 deg. form] is
related to: 2 =[sqrt(5) +1]/ Φ, as 60 deg. =Arcos[1/2].
Since: θ = Arctan[sqrt[Tan [φ]]], i.e angle θ is related to angle φ, and
Since:
Sqrt[ [tan(45)+sqrt[5*tan(45)]*cos(60)] =tan(51.82729238…)=1.27201965..=T =sqrt[Φ] = tan[ θ], then
angles θ, and 45, are related also to angle 60 deg.
LOGARITHMIC RELATIONSHIPS by PANAGIOTIS STEFANIDES:
According to my manipulation of basic logarithms [Using the known relationship for finding logarithm to base 2 of a NUMBER as the ratio : Ln[NUMBER] / Ln[2] , Ln being the Natural Logarithm [base e], all [positive Real] numbers [ or angles ] are related to angles Arctan[T] =51.82729237..deg., and 45 deg:
[tan(45)+[sqrt(5)+1] / e^{[ln[ln[NUMBER]] ]- ln[ln[sqrt[NUMBER]]]}]^0.25 = = 1.27201965..=T =sqrt[Φ].
Examples:
[tan(45)+[sqrt(5)+1] / e^{[ln[ln[2]] ]- ln[ln[sqrt[2]]]}]^0.25 = 1.27201965..=T =sqrt[Φ]
[tan(45)+[sqrt(5)+1] / e^{[ln[ln[0.1]] ]- ln[ln[sqrt[0.1]]]}]^0.25= 1.27201965..
Number: 2=2*tan(45)=1/[cos(60)], Logarithmic Relationship:
The Logarithmic form involving number 2, [giving the relationship for the 30/60 deg. orthogonal triangle, from which the equilateral triangle is produced], is as follows:
Ln[Ln[+positive real NUMBER or angle]] – Ln[Ln[sqrt(+positive real NUMBER or angle )]] = =0.693147181..
Ln(2) = 0.693147181..=Ln[1/[cos(60)]], = Ln [[ sqrt(5*tan(45) +tan(45)]/T^2].
e^[0.693147181.]=2
Example:
Ln[Ln[16]] – Ln[Ln[sqrt(16)]] = Ln(2)= Ln [[ (5*tan(45) +tan(45)]/T^2] = 0.693147181...
A GREAT PYRAMID MODEL [G.P.] RELATIONSHIP WITH THE 5 SOLIDS:
A Great Pyramid Model [G.P.], its structure being based upon this theory of the proposed new form of the “Most Beautiful Triangle” relates θ and 45 angles triangles with angles φ, found on each of the isosceli triangular faces of the G.P.
By cutting in half one of this face [by a vertical from its vertex perpendicular to its base] we get two orthogonal triangles, bearing angle φ [and 90- φ].
Adding them along their hypotenuse we get a parallelogramme plane.
Placing three such planes each perpendicular to each other, automatically we get 20 equilateral triangles[60 deg.] defined by the vertices of the such structured planes.
So we constructed the icosahedron via this process.
In a similar way we do so for the structure of Ether [Quintessence] or dodecahedron [ since we find Φ^2 = 2.618033989..or T^4, in a selective section of this structure, and , also angle φ].
The 45 deg. [ Arcsin[1/sqrt(2) = Arcos [1/sqrt(2)] structures the square and the cube, which , within cube relates to sqrt(3), i.e. angle: arcsin[1/sqrt(3)]=35.26438968..deg. and 90-35.26438968.. = 54.73561032..deg., both of which found within selective sections of the tetrahedron and the octahedron.
Theory of Lines Building the World
Solids are formed by triangle surfaces originating from the two triangles the special scalene orthogonal and the isosceles orthogonal.
Triangles sides are formed from lines based on kinds of 4 lines in geometric ratio to each other T, i.e. lengths corresponding to : T^3, T^2, T^1 and 1.
The special scalene orthogonal is formed from the lines T^3, T^2, T^1, the isosceles orthogonal from 1, forming the square with sides 1.
A second triangle similar to the special scalene is formed with sides T^2, T^1,
and 1. This is constituent orthogonal triangle of the first.
Two pairs of such triangles joined together build[SOMATOIDES] the first solid constituent of a G.P. Model and thus of the 5 Platonic or Euclidean Solids.
http://www.stefanides.gr/html/All_triangles_derive.htm
Further, relating to the two Timaeic Triangles, I would like to elaborate on my proposed process of structuring Solids:
ROOT GEOMETRIES STRUCTURING THE ICOSAHEDRON, THE OTHER BASIC POLYHEDRA AND THE GEOMETRIC FORMS RELATED TO THEM
http://www.stefanides.gr/html/Root_Geometries.htm
Under Root Geometries Structuring the Icosahedron, the other Basic Polyhedra and the Geometric Forms Related to Them, we refer to the basic geometric configurations which, as this theory contemplates, are necessary for the progressive mode of formation of the five polyhedra, via lines, areas and volumes.
The Root Geometries are two configurations of triangles. A Special one, the Quadrature Scalene Orthogonal Triangle [Author’s interpretation of the Timaeic Most Beautiful Triangle] with sides [T^3], [T^2] and [T^1] in geometric ratio T, which is the square root of the golden ratio[ Φ], and the Isosceles Orthogonal Triangle, with its equal sides [T]. The surface areas of these triangles are taken perpendicular to each other and in such, naturally, defining an X, Y, and Z system of coordinate axes. In so, the coordinates of the first are [0,0,0,], [0,0,T^2], [T,0,0] in the X-Z plane, and those of the second are [0,0,0,], [T,0,0], [0,T,0] in the X-Y plane. A line from [0,T,0] to [0,0,T^2],creates the same Scalene Triangle in the Y, Z plane.
Arctan [T] is the Scalene angle [θ] of the Special Triangle with the property that the product of its small side by its hypotenuse is equal to the square of its bigger side: [T^1]*[T^3] equal [T^2]^2 [Quadrature].
Using a pair of the Special Scalene Triangle, and a pair of a Similar Kind of Triangle [Constituent of the Special] with sides 1,T and T^2 [Kepler's (Magirus) Triangle with sides 1, sqrt(Φ), and Φ] a Tetrahedron [dicta Form 1] is obtained, by appropriately joining the edges of the four triangles, with coordinates: [0,0,0,], [0,0,T^2], [T,0,0] and [0, 1/T, 1/T^2].
By joining, a line, from point [0,T,0] to point [T,0,0], a Second Tetrahedron [dicta Form 2] is obtained [ as a natural extension of Form 1], with co-ordinates: [0,0,0,], [T,0,0],[0,T,0] and [0,1/T, 1/T^2], having as base, on the X-Y plane, the Isosceles Orthogonal Triangle mentioned above, with coordinates[0,0,0], [T,0,0] and [0,T,0].
Doubling this triangle, in the X-Y plane, a square is obtained of side [T], with coordinates [T,T,0], [T,0,0], [0,0,0], and [0,T,0].
By connecting a line from point [T,T,0] to point [0,0,T^2] a third Tetrahedron [dicta Form 3] is obtained with coordinates:[T,T,0],[T,0,0], [0,T,0] and [0,0,T^2],
having also as base, the Isosceles Orthogonal Triangle with same dimensions [mirror image] as that of [Form 2]. The three Forms are wedged firmly together, leaving no empty space between them. Their volume ratios
Form 3: Form 1: Form 2 equal to [1/6]*[T*T*T^2]: [1/6]*[1*T*T]: [1/6]*[T*T*(1/T^2)] is the golden ratio [T^2], and the sum of volumes of Form 1 and Form 2 equals to [1/6]*[1*T*T ] + [1/6]*[T*T*(1/T^2)] equals to [1/6]*[T^2+1] equal [1/6]*T^4[SINCE T^4-T^2-1=0],
the volume of Form 3.The volumes of the three Forms sum up to [(2/6)T^4 equal to (1/3)T^4].
Two of the four bases of Form 3, are symmetrical orthogonal triangles, with coordinates [T,0,0], [T,T,0], [0,0,T^2] and [T,T,0], [0,T,0], [0,0,T^2], each of which has an angle [ φ], equal to arctan[T^2].
Two such triangles joined coplanarily and symmetrically along their bigger vertical sides, create one of the four triangular faces of a great pyramid model with coordinates [T,T,0], [0,0,T^2] and [T,(-T),0].
The Structure of the three Forms bound together [dicta Form 4] with Volume [1/3]*T^4 is one quarter of the volume of the great pyramid model, which has a square base of side 2T, height T^2 and Volume [4/3]*T^4.
Splitting one of this model’s triangular face into two orthogonal co-planar triangles to form a parallelogramme [with sides T^1 and T^3], we have constructed the basic skeleton of the Icosahedron, since three such parallgrammes, orthogonal to each other, determine its twenty equilateral bases, by joining adjacent corners in groups of three, by lines.
Similarly, we proceed to the construction of the dodecahedron, the tetrahedron, the octahedron and the cube, together with their related forms such as squares, circles, triangles, circumscribed circles to the parallelogrammes of the polyhedra skeletons, circumscribed spheres and logarithmic spirals.
Reversing the whole process, the volumes decompose to the areas of the triangle surfaces structuring them which, in turn, resolve to four line traces harmonically codified in space.
An more illustrative related Link may be searched under :
http://www.stefanides.gr/html/Root_Geometries.htm
© Copyright 1986 - 2012 Panagiotis Chr. Stefanides
Panagiotis Stefanides
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http://www.stefanides.gr/pdf/PROPOSED_GEOMETRY_OF_THE_PLATONIC_TIMAEUS_GREEK.pdf.pdf
http://www.stefanides.gr/pdf/BOOK%20_GRSOGF.pdf
http://www.stefanides.gr/Html/Photo.htm
http://www.stefanides.gr/pdf/PROPOSED_GEOMETRY_OF_THE_PLATONIC_TIMAEUS_GREEK.pdf.pdf
Ref:
https://www.researchgate.net/project/Philosophy_II/bench/Timaeic_Platos_Triangles/?ev=proj_det_bnch_nam
Regards from Athens,
Panagiotis stefanides
Panagiotis,the above are somehow too technical for some of us.Is it possible to give a summary of your conclusions/hypotheses/interpretations in plain English without the mathematics?
Dear Cornelius,
I thank you.
I give following an abstract:
ABSTRACT
Proposed Interpretation of the Timaeic Plato
This work, proposes a different interpretation of Plato’s Most Beautiful Triangle.
In section 53/54, of PLATO'S "TIMAEUS", PLATO speaks about the triangular shapes of the Four Elemental Bodies, of their kinds and their combinations:
These Bodies are the Fire (Tetrahedron) the Earth (Cube), the Water (Icosahedron), and the Air (Octahedron). These are bodies and have depth. The depth necessarily, contains the flat surface and the perpendicular to this surface is a side of a triangle and all the triangles are generated by two kinds of orthogonal triangles: the "ISOSCELES" Orthogonal and the "SCALΕΝΕ" Orthogonal. From the two kinds of triangles the "Isosceles" Orthogonal has one nature. (i.e. one rectangular angle and two acute angles of 45 degrees), whereas the "scalene" has infinite (i.e. it has one rectangular angle and two acute angles of variable values having, these two acute angles, the sum of 90 degrees). From these infinite natures we choose one triangle "THE MOST BEAUTIFUL".
Thus, from the many triangles, we accept that there is one of them "THE MOST BEAUTIFUL.
Let us choose then, two triangles, which are the basis of constructing the Fire and the other Bodies : "Το μεν ισοσκελές, το δε τριπλήν κατά δύναμιν έχον της ελάττονος την μείζω πλευράν αεί."
My Proposed New Interpretation:
One of these two is the "ISOSCELES" orthogonal triangle, the other is the "SCALENE" orthogonal triangle, its hypotenuse having a value equal to the "CUBE" of the value of its horizontal smaller side and having its vertical bigger side the value of the "SQUARE" of its smaller horizontal side. The value of the smaller horizontal side is equal to the square root of the Golden Number, the ratio of the sides is equal, again, to the Square Root of the Golden Number (geometrical ratio) and the Tangent of the angle between the hypotenuse and the smaller horizontal side is also equal to the Square Root of the Golden Number (Θ =51 49-38-15-9-17-19-54-37-26-24-0 degrees). The product of the smaller horizontal side and that of the hypotenuse is equal to the "SQUARE" of the bigger vertical side, and the following equation holds:
T^4-T^2-1=0 , T = SQRT [(SQRT.(5) + 1)/2].
N.B. Kepler [ Magirus] Triangle is a similar one but not the same:
http://en.wikipedia.org/wiki/Kepler_triangle
Regards,
Panagiotis Stefanides
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http://www.stefanides.gr
Very immersive discussions! Coming on board rather late...The history of science and its philosophy has in large part been a history of instrumentation. Whether it be the chronology of successive technological advancements or the rise and fall of systems of thought.And as such, though the object of investigation may be real (ontologically-speaking), the meaning, learning and knowledge through the course of interacting with it is always obscured- "through an instrument dimly." Thanks in part to the human mind as an instrument, and also the collective consciousness as a ferment for ideas, or a graveyard of lost/redundant ideas. The finite does not attain to the universal, as Gödel admitted with his attempts at axiomatizing math.Yet within the bounded is to be found the boundless.Even models and metaphors can be beautiful with much depth of hope. As one might be awed by the implications from Cantor's dust.
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