Good question.

The answer is that the higher dimensions are more real than our experience isolated to the third dimension. But to understand why this is the case is difficult. In order to understand it you have to rethink your view of the world somewhat. In our worldview Logos/Physus is the most basic duality. But between them is the Nomos (Order, Math) which allows theory to connect to experimental results. All good scientific theory mimics math in operationalized words that reflect how phenomena works. Without all three you could not have science. But Nomos being nondual comes before (is a priori to) the split between Logos/Physus. How do we know this? because if the universe did not have anything in spacetime there would still be the math. And the math has some very odd characteristics and those actually drive our world and constrain it in specific ways that show up in physical phenomena. The simple answer is that Logos and Physus describe Being and the Nomos describes existence as it looks from the Physus/Logos split. Being is in fact a unique Indo-European linguistic kink and thus is not universal in any stretch of the imagination, except by fiat of killing off many of the other peoples and worlds, and languages that did not have Being. So that means that nomos is nondual and prior to the rest of nature that appears when there is something in spacetime but it has structure even if nothing ever articulates that structure. Part of that structure is N-dimensions. But N-dimensions though infinite are not all the same. In fact the fourth dimension which is the dimension that we really live in has some pretty unique properties compared with the other dimensions. But dimensions are different from each other. And we see this best in the size of hyperspheres which get bigger then tail off into the infinitely small. The peak of size of hypersperes is around the 5th to the 7th dimensions. So although they are infinite this does not mean that the structures that are interesting go on forever, but rather oddly the interesting structures all exist at the lower dimensions, near where we are. The oddities like the hypercomplex algebras drive a lot of structure in the low dimensions. Also the fact that the fourth dimension uniquely has no set topology is important. So since Math constrains the physics it must be real, in fact in some sense more real than the physics, because those constraints would be there even if the matter were missing.

There is more to this answer than what I have mentioned but this is a good place to start to answer the question of why mathematical dimensions are real, more real than the phenomena that fills spacetime that the math constrains. The math has to constrain it otherwise science could not rely on mathematics to guide its theorizing about phenomena.

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How do we know that mathematics (Nomos) is prior to phusis (Nature, Physics)?

(responding to a comment from Chase Quinton)

What you have to grasp to understand this idea is basically the essence of Kantian Philosophy. What we need to see really how deep the Kantian picture is in order to accept this idea. Basically he worked out an answer to Hume’s Skepticism. But more than that what has come out of Kant’s approach is the very basis of our scientific endeavors since his time. In effect no one has figured out how to surpass the limits he set on reason, nor the way that he has tied reason to experience to produce understanding, or the way he has managed to rescue causality from the mire of mere sequences of events. In other words Kant’s view has become our predominant view in our worldview concerning the efficacy of Science. And that view demands Transcendental Idealism as the route to Transcendental Realism. Everything else has gaps that cannot be explained. Kant got rid of the gaps, by saying first we project spacetime which we understand through math, and then we project the categories by which we get objects within spacetime and causality that may be objective, and the only way to know what is real is through these projections that we make unconsciously. What it does is by basing everything on projections is to make sure there is no gaps between us and what is real. Those projections we make are the reality, but that does not mean we are constructing that reality we see, but it means that what comes through the medium of the projections IS real, as real as anything gets. All other views have unexplainable gaps between reality and our attempts to be objective as subjects, and cannot explain things like causality on which Science is based.

Now since Kant philosophers have tried every possible way out of this box that Kant put us in. But they all fail one way or another and if you look at both Continental and Analytical philosophers they all accept and harken back to Kant one way or another. Because without his Copernican Turn toward critical philosophy by weakening reason and tying it to experience to produce understanding there is no way to explain how science actually works. All the other opinions are just tweaks to this basic positions or reactions that merely entrench it further. In other words we know that nomos comes before physus because logically spacetime has to come before anything can be in it. And Nomos is how we understand spacetime. And we can understand it become it comes out of us a a projection prior to our experience. Then we know that objects that are physical come next because things in space logically have to come after the place that they are in, and we know them because we are projecting the categories after spacetime. And we know these things because of our judgments that are aligned to the projections, and the judgments are in language. We project what we can then understand and interpret as judgments. Essentially this means that objects have to adapt to us and our projections rather than us adapting to them. We can only know what we are prepared in advance to know, we cannot know anything that we are not prepared to know prior to the knowing by our projections. Everything I am telling you is explained by Bernstein in his lectures of Kants at http://bersteintapes.com. So if you want to know more then listen to those tapes. We are projecting synthetic a priories, because that is the only way we can know anything. Whatever we know is already part of us from the beginning. And because of that what we know via our prior projections is the only thing that is Real to us.

It is like a current author (of Is that a Fish in your Ear) on translation said at the LA Times Bookfair recently. If we sent people off into space and they met aliens and learned their language and spoke to them, and they came back we would ask them what the Aliens said. If they said we would like to tell you but their language is untranslatable, we would doubt their veracity. By its very nature if they learned the language of the aliens and spoke to them then somehow what the aliens said would be translatable into something. This I think is another version of Kant’s argument. The very nature of language is that it is translatable into any other language somehow, to some degree, with some fidelity, perhaps not accurate or correct but to be language at all it must be translatable. Kant is merely saying the same sort of things about nature. Nature has to be experienceable. To be experienceable there are certain rules that it must adhere to and if the phenomena does not adhere to those rules governing our experience, we cannot experience it, and those rules come from inside us, not from outside. We may not be able to experience everything, what is real to us is what we are capable of experiencing and understanding. We use reason to aid in understanding. Left to its own devices reason makes up fantasies about the world, but with respect to interpreting phenomena by the rules of judgment to produce understandable experience reason does a good job and giving us access to the realities we can handle. Elliot’s quip about this is that Human beings cannot take too much reality, but the reality we can take is as real as anything gets.

I don’t want you to get confused about whose idea this is. It is Kant’s idea. Please argue with him if you must. People have been arguing with him ever since he writes the First Critique, but no one has found a way around this argument. You cannot accept it, in which case Science as we know it breaks down for you, or you can accept it in which case Science sorta works, and just enough to discover amazing things, but not enough to tell you how real the things are that they have discovered. So you have to kind of decide if you want Science to appear to work at least half way OK, or not at all. For Kant there are no noumena. He only posited noumena so he could show that this is an empty concept. Noumena do not exist because everything we experience via our projections is itself real, as real as anything can be. if there were nounena, i.e. realities that we cannot know, we would not know them, and so speaking about them would be a moot point.

This is not Platonic. For Plato and other metaphysicians before Kant there are unknowables, and the metaphysicians from their privileged position will tell you about them. But Kant is going to say every time to these in his critical philosophy, if it did not come from me ultimately then I cannot know it, and so it is irrelevant. I would rather than a science that told me novel things about the world that are interesting that I can know, that myriad unknowable things no matter how fascinating. If you want to know a little that is real about the universe then Kant is the only show in town. If you want to know a lot that is not knowable about it then there are other metaphysical paths to follow. There are a lot of problems with Kant’s thinking but on the essential point of the lack of value of ungrounded metaphysics there is unanimity. Un-grounded here means not grounded in our nature as human beings, and tied down to our experience, such that it produces understanding of the world. Another way to say this is what is real is what we can understand, and things e cannot understand ultimately because they go against our reason are not real, and are worthless to us.

I hope this helps.

Most art is self indulgent. It is produced by the self to express the self either to the self or others. So somehow if the artist is not indulging the self then then there is little to express. I would say that indulgent art is allopoietic in the sense that it is producing something other than one’s authentic self. But good art is autopoietic in the sense that it is genuinely producing the self qua self which is the unique individuation of the artist. But of course this is just a guess. What it means to be  “self” indulgent is open to interpretation. But art qua art is an expression of the self no matter how abstract or minimalistic or conceptual we attempt to be. Good art is not just a self-expression but also is archetypal in some sense. As Kant says Beauty is intersubjective in some sense. Great art hits a chord we all can appreciate somehow, or at least ought to appreciate to the extent we realize our humanity within ourselves.

To me the art of Bacon that Deleuze praises is self-indulgent. I don’t see what Deleuze sees in it.

But for instance the statue of Laocoon cannot ever be called Self-indulgent because the self that expressed it expressed purely the human condition. The self of the artist is effaced in its immersion in the human condition. But by that it does not lose its individuality, but instead heightens it to embrace everyone somehow  which is what Jung calls the individuation of the self.

What is the use of listing names? It is kinda silly since we can just go to wikipedia and get those lists. And Nine, forget any specific number because there have been thousands, but the some of the names listed in other answers are significant. The question is what is it that makes a thinker on logic’s work influential. What are the criteria? The main criteria is that they should profoundly change how we think about logic. Best example is Peirce. He has probably done more to change the way we think about logic than anyone else since Aristotle who created it for us in the West. But there has been so much work in logic over the last century that it is difficult to say who is most important. Fuzzy Logic is definitely important. Para-consistency of Priest is important. But perhaps the most important is G. Spencer Brown who along with Bricken and Hellerstien and Kauffman have created a boundary logic that is the logic of Masses to rival the Syllogistic Logic of Sets. But then also very important is August Stern and his Matrix Logic. There are now myriad deviant logics including Quantum Logic that are significant. Another major contribution was Higher Logical Type Theory of Russell. But also Wittgenstein’s Tractatus and his later retraction Philosophical Investigations and Philosophical Grammar drew a lot of attention to logic. Also there is Topos theory which is about the Categorical representation of logic. There has been a lot happening in Logic over the last century. Logic is not what you think it is anymore. It is a whole field with multiple profound contributions. No standard list of names is going to capture that reality. A renaissance in Logic has occurred over the last century and everyone basically missed it. But eventually it will change the way we think about everything. An important contribution to this was the book Life Itself by Robert Rosen where he shows that causal and inference structures can be analyzed categorically into entailment structures and that these are more complex than we bargained for, complex enough to let biology into the science that includes physics without appeal to vitalism. Lots going on there to learn about, think about and use as a tool for making our theorizing better.

Easy to list them, but hard to say why they are influential. They are influential because they are exploring the ideas of Heidegger based on those of Husserl, Hegel and Aristotle. They have rediscovered the meta-levels of Being. They have explored this territory discovering

• Pure Being – Husserl
• Process Being — Heidegger
• Hyper Being — Heidegger, Merleau-Ponty, Derrida
• Wild Being — Merleau-Ponty, Deleuze, Castoradis
• Ultra Being — Zizek, Badiou

If you can understand these kinds of Being you can understand the nature of the essential nature of the Western worldview.

Zizek is the most influential because he is like a pop star of philosophy due to the fact that he has an opinion on everything. But he also has some deep points to make especially when you contrast him with Badiou and understand his reduction of Lacan to Hegel. He is appealing because he has interesting things to say about everything under the sun. He likes to say things that are controversial and thrives on intellectual jousting with the other intellectual pop stars of our time.

Probably Zizek’s books particularly Parallax View but the other significant ones as well all work together to give an interesting reevaluation of Hegel now that Marxism is free from the Soviet Communist yoke. The first such book was Critique of Dialectical Reason by Sartre. Recent books by Fredrick James are also offering reassessment of dialectics in the face of Postmodernism. But Zizek has actually via Lacan come up with a genuinely new view of Hegel and if taken seriously that is going to make a difference in the way political philosophy is understood. The first big break was the political economics of Bataille with the idea of the Accursed Share. We can see the effects of that in Deleuze and Baudrillard. But Zizek uses Lacan and Hegel to drive the point home with a vengeance which should transform the way we think about politics fundamentally in terms of the blatant Ideology of our supposedly post-ideological age after a century of ideological warfare. Just because you have beaten all ideological foes does not mean Ideology has vanished, it has just gone underground because there are no other standing ideologies to compete any longer. Zizek makes the point that unconscious ideology is probably worse than blatant ideological discourse because everyone thinks they are free from ideology.

What is the nature and structure of the Western Worldview. And beyond that what is the nature of Scientific Discovery within our worldview and what are the alternatives to that which might shed some light on its underlying assumptions that are not immediately obvious. And beyond that what can be done with respect to transforming this worldview given its inherent structure to make it so that our worldview which has become dominant and global does not destroy the planet, other species, and ourselves.

This is my problematic. I have been pursuing research in this problematic for years. If you want knowledge you have to have a problematic. And it is that problematic that gives rise to the significant and relevant questions that need answers in order for you to gain knowledge with respect to the subjects that you find fascinating. There is no magic bullet that makes knowledge suddenly appear out of nowhere. Look at the people who have contributed to our knowledge within our worldview, they worked hard, very hard for long long time before they figured out what ever their contribution to our knowledge happened to be. Many tried and many failed before someone discovered something of value that we pass on in this tradition. So there is no magic bullet when it comes to seeking knowledge, but only hard work, probably without reward, for a lifetime and then results are not guaranteed.

One of my favorite books is Knowledge Painfully Acquired by Lo Chen Shun who thought about the relation between Confucianism and Buddhism for years when Chinese thought was under siege and tried to formulate in his mind a simple way of understanding the uniquely Chinese contribution that Buddhism had no answer to, and so his book represents a life time of deep thought about the key points of his native culture that were under threat, and why they were more valuable than what Buddhism had to offer.

We need this kind of thought very badly today about our tradition. What is the key thing about our tradition that if changed would prevent it from destroying us and the world too, along with all other species. What is going to have to change fundamentally so that the earth does not become like Venus too hot for our kind of life. Not enough thought has been done concerning this issue. So I decided long ago to work on it. And I have come up with some results that appear in my various papers. But the results are not good enough to avert disaster. And not enough others are thinking deeply about it to make a difference. But I am trying and you should be trying. If we all try very hard then perhaps someone will figure out how to cut the Gordian Knot that binds us to the fate we seem to have decreed for our planet which is against our own interests.

Pick a problematic of your own. Work on what fascinates you. Attempt to find the cutting edge of your discipline and do something creative there that advances the tradition. The success is in the working toward the end. It is human success. Success at being who we are inquisitive animals on a lonely planet with no other refuge in sight, yet we are fouling our own nest. What is the root of this perversion in our souls. What makes us suicidal. It is one thing we abhor that anyone would become a suicide bomber taking others with them in their evil self-destruction. But what do we say to a species that want to take all other species with them in a similar senseless self-destruction through the destruction of the planet, and every living thing on it including ourselves.

The situation is absurd. But if we don’t strive to understand what is driving us toward that self-made apocalypse then there is no chance we will change the direction of the car careening off the cliff. We have to stop it or turn it aside before it goes over the cliff edge. Regrets later will not be enough to satisfy the generations of suffering or dead or forever unborn victims of our insanity.

My favorite method in this regard is Heuristic Research by Clark Moustakas

http://www.cosmicplay.net/method/extra/extraheur1.html

but he also wrote a general book on Phenomenological Research Methods
http://books.google.com/books/about/Phenomenological_Research_Methods.html?id=QiXJSszx7-8C

This is an interesting and deep question.

There are actually three things to be concerned with:

• Logic as the core  of Language
• Various possible foundations of Mathematics
• Mathematics proper.

The basic answer is that no one knows what the relations between these things are. Does language or numbers come first, and then what is the relation between the logical core of language and the foundations of mathematics, and finally what is the relation of the foundations to all forms of mathematics which seems to be expanding very rapidly over the last few centuries but almost exponentially in the last century.

I have written quite a bit about this question in response to Badiou’s Being and Event where he claims that Ontology is Set Theory. There are a lot of problems with this position, but something it forces us to do is take seriously the relation of Being to Set Theory and other foundations of Mathematics which is something we are not forced into. Mainly this is because the Ontologists don’t know enough math and mathematicians do not think about ontology. But Badiou prides himself on having gotten the mathematics down well enough to philosophize about it. What is wonderful about that is that Badiou uses Cohen’s work and generalizes it to give it an ontological bearing. And that helped me because at the time I was trying to figure out what the next level up of abstraction beyond General Schemas Theory and wondering if I needed to define that in order to ground General Schemas Theory. Basically using Cohen’s result in Set theory that says that there is some independence between certain axioms and the others, we can generalize this method and see that if something does not matter at the level of abstraction you are at if you changed it, i.e. if you cannot tell the difference if it changes, then it is not a transcendental constraint that you have to worry about at that level, and that frees us from being concerned with upper level transcendentals if they have no lower level effects if they are changed. This is an excellent point. So if changing the worldview structure did not change anything at the General Schemas Theory level then I did not have to worry about grounding General Schemas Theory in a higher theory of the structure of the worldview. And I decided that was the case so it freed me of a lot of work that would have been probably unnecessary.

But what you see in Badiou is the focus on Set theory as the basis for mathematics when there are actually several contenders like Category Theory (my personal favorite) and Mereotopology, etc. Seems to me we have to consider all the various possible foundations of mathematics and their intrinsic variety and then generalize from that. Badiou wrote a companion volume to Being and Event which I have not read yet but it applies Logic to Worlds. This seems implausible but we will have to wait to read it to criticize it.

But my view is this which I have formulated on my website at http://holonomic.net. Logos and Physus are the most important duals in the Western worldview, and Nomos is the non-dual interface between them. Logic is the Core of Language (the Physus of the Logos) and thus we can see that schematization is the core of the Physus (the Logos of the Physus as what Kant calls an A priori synthetic projection of order as the singular of spacetime). Thus Logic appertains to Language (the Logos) and its opposite is schematization which structures both language and also our apprehension of the intelligibility of organized things in spacetime. But the nomos is separate and that is where the math is. And by the way it is only Indo-European languages that have Being, so it is a special anomalous case. So to identify it with set theory is obviously wrong. It seems to me that mathematics comes first because Nomos is the nondual that is there before Logos and Physus come into existence and separate from each other as a duality. And we can represent logics as Topoi (the category for logics) and that means that we can have a purely mathematical view of the structure of logic prior to the existence of Logic as the core of Language.

However, once Logic itself exists then it plays over the various possible foundations of mathematics that make up our meta-mathematics. Meta-mathematics comes back from Logic toward Math and tries to understand its foundations, and so both Logic and Math each help us to understand the other. Both make sense of each other by providing a different context in which the other can be explored.

However, it is not that Logic and Phusis come from Mathematics as much as they are pre-constrained by mathematics. For instance, in Science we look for mathematical foundations of our theories and then look for anomalies that disprove our theories that are structured based on the mathematics. Theories that withstand this kind of disproof tend to have a very close operationalization through their connection with the math. Similarly, there are myriad logics, and understanding those logics and their differences help us to find different ways to interpret the math and to see the possible foundation for the math, and it helps us structure our theories better so that they are logically consistent, complete, clear, verifiable, validatable, and coherent.

Hope this helps to clarify the mutual elucidation of Math and Logic which happens in the context of philosophy of Science and in the pursuit of science. This is really at the core of our worldview. And it is something we really need to understand, and don’t completely understand as yet.

Something you have to understand about Hegelian dialectics is that there is two forms. One is what is in the Phenomenology of the Sprit/Ghost/Mind and the other is the formalized structure in the Logic. So what is happening in the phenomenology of mind is not exactly formal, not exactly perfectly dialectical in terms of format. So what is in the Phenomenology of Geist is so much more interesting than the later formalize of ‘Dialectics’ that we get in the Logic. It is clear that History does not follow any formal rule. So the idea that History is dialectical in the formal sense clearly wrong even if we can get approximations when we project it onto history. But the idea of History being dialectical in the looser sense is still of some interest. This is because the idea is that at a given time in intellectual history, all the possible permutations appear and ultimately cancel each other out giving rise to a deeper view, and then the same thing happens with the deeper view. The perfect example of that is the pre-Kantians and their dogmatic philosophies prior to Kant’s Copernican turn. Kant’s transformation of Philosophy has determined the direction of philosophy ever since. And if you look at the commentaries on Kant and all the post-Kantian philosophies then you can see that every possible interpretation of Kant has been tried out, without anyone escaping the limits that he set on reason that ground Science even to this day. He only wanted to ground Newtonian science, but ended up defining Science itself from his time till now. But there have been within this space he opened up some crucial philosophical developments within that conceptual space as philosophers grappled with the fundamental philosophical problematic he set up. So if we look at the different ideas and their relation to each other what we notice is that over time as the problematic develops all the permutations tend to cancel each other out, and eventually someone will come up with a fundamental transformation that will reset the playing field, and start the dialectical process all over again. In the formalized view given by Hegel later the driver of the process became negation, but the real driver is the fundamental human nature that produces variety, and the permutational exploration of the space of possible Science grounding philosophies. And it is possible to see when we study the space within the problematic that the various views are arrayed antithetically to each other in such a way that they ten to balance each other out. And this is natural phenomena according to Hegel with his idea of Absolute Reason. Absolute reason actually is just science applied to History and says that everything in history because it is in the universe governed by physical laws has a reason. So if you look at any phenomena closely enough you will see absolute reason playing itself out in all phenomena. Historical phenomena is just a horizon that is infinite and so it is turtles all the way down, i.e. there is reason all the way down, natural laws apply to everything and so whatever you look at no matter how detailed and minute has a reason. And part of that reason is the natural variation of our interpretations of history and events on which there are multiple perspectives that tend to cancel themselves out. When we realize this cancellation that comes from the human intellect producing antimonies we can see that it is superseded by a different view that is deeper and then the antinomies play themselves out at the next deeper level of understanding or comprehension of the phenomena.

Hegel justifies his view by saying we live in a law governed universe and that the natural laws apply to everything and thus all things have sufficient reason for their existence from the laws of nature. But just because we can say that in general, and we have even more reason to say that now, does not mean we know the reason, that is a speculative activity to produce interpretations that contain those reasons. The dialectics have more to do with our ways of seeing the reasons and our interpretation than the actual mechanism of history itself. For instance if a process is random the that is the reason it  is the way it is. But there are different interpretations of what randomness is and those tend to cancel each other out. What drives history is natural laws operating and emergent levels that articulate phenomena. When we look at that phenomena we know it is driven somehow by natural law at given emergent levels. But our different views of the phenomena tend to cancel each other out unless there is a symmetry breaking in which some new view supersedes all the others. Then we get the same variety production of view until a new symmetry breaking occurs. The succession of views based on the cancellation of interpretations and permutations of ideas is the expression of the dialectical development in the history of ideas. The phenomena does not necessarily change but our interpretations of it can change radically. Thinking this happens though negation only is just a reductionist and reified view of a interesting and dynamic process by which our understanding of the natural world and ourselves improves over time by a lot of hard work by a lot of people toiling at the business of science over the centuries. The justification is the lawfulness of nature. The workings of absolute reason as a reflexive self-consciousness about history can be seen in every field in which myriad theories are produced until one turns out to be either more interesting or better at explaining facts than others, and then we readjust and begin building from there. This process is absolute reason working itself out on history of consciousness and applied to a world in which everything is determined in some way by physical laws that hem in our expression of inner freedom.

[This answer needs improvement 2012.08.15 kdp]