A Mind for Madness

Musings on art, philosophy, mathematics, and physics


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Table some issues

Now that I’m done with undergrad, I’ve decided to throw out there some of the things I’ve attempted to have answered, but people always seemed to beat around the bush. Maybe my readers (which I’ve calculated to be somewhere between three and five readers) can help.

1. Setting: Sophomore year: Eastern religions. My prof is amazing and a devout Buddhist. He claims that the religion is one of the few perfectly internally consistent religions. So I ask, how can it be consistent that the principle teaching is that the world is an illusion, and another primary teaching is that you shouldn’t harm another living being? If you truly believe the world is an illusion, then harming someone is an illusion, thus cannot be bad. I was given some mumbo-jumbo about karma, but I replied that this assumes you are putting negativity out while you harm this living thing. This went on, but I’m convinced that it is possible to harm something without feeling hatred etc, he claims that this is still against the teachings. Is this a contradiction? (I think yes).

2. Setting: Recurrent throughout all four years. Would independent art still be considered good if a large mainstream audience started taking interest? So I should elaborate, I guess. Often times it seems like small independent artists that create extremely challenging art are considered “true artists” while mainstream people are considered “pop artists” or “money makers.” It seems to me that even if the art doesn’t change, a sudden burst into fame can cause the art-lovers to despise this artist. As if it is the struggling artist that is the cause of good art and not the art itself that is under consideration. I guess a good comparison would be the filmmakers David Lynch vs Spielberg (although Lynch has had some commercial success).

3. Setting: Senior year: 20th Century philosophy. Is empathy or language epistemically primary? (I won’t even go there). Okay. Yes I will. Suppose we find a “savage” that has been in the woods on their own since birth (i.e. they have no language and have never encountered another human being). I claim that if they saw someone crying, they would remember when they cried and understand what that meant (i.e. the empathy transcends language and is thus epistemically primary). My prof claims that without language the “savage” wouldn’t have memories, or at least wouldn’t be able to access memories of when they cried (i.e. language is epistemically primary). This seems to be a question more directed at cognitive scientists than philosophers though, since it seems as if this could be empirically tested quite easily.

4. Setting: Probably junior year conversation with brother. Why is science fiction as a genre not considered serious as an artistic genre when there are so many examples that show that it is? E.g. Dhalgren, Slaughter-House Five, The Fountain, The Ender Quartet, etc.

5. Setting: Trying to get into grad school (also came up in an ethics class discussion). Is there any fair way to measure ability other than a standardized test? The GRE is a horrid measure of ability, but I just can’t think of any other way to do it that would be more fair (does the concept of “more fair” exist?).

I actually had more when I decided to write this (and some of these weren’t on my list), but once I got through the first one, I couldn’t remember any of the ones that I originally thought of. Maybe I’ll add more later.


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Wittgenstein

Okay. I admit it. *Don’t hurt me.* I don’t own a copy of Wittgenstein’s Philosophical Investigations. Yesterday I decided it was time to remedy that. I have now come to the conclusion that no one owns it. The cheapest copy I could find at Barnes and Noble was $40. How absurd is that? It consists of 100 aphorisms. That isn’t even 100 pages. And it was paperback.

Well, enough of my ranting. Needless to say, I still don’t own a copy of it (and neither does the entire public library system in my district…what is this world coming to?). At least I found a wonderful website to keep me occupied until I find a copy here.

Some comments are necessary. I always seem to forget about the first half of this text. Everyone remembers the “language is everything, ” but I seem to forget the evolution of language and meaning as function parts. So Wittgenstein rejects the old view that language is stagnant and made up of words that have fixed meanings (look in a dictionary). He talks about how it is sort of living in that things get added, meanings change, etc. It evolves.

Why is this important to me? Well, mathematics is a language. Here is a plug for the position I’ve advocated several times in the past on this blog. Since math is a language it is not fixed or set in stone! People don’t seem to get that. Maybe the more natural definition to work with is monoid and not group. Maybe the more natural construction is category and not set. Mathematicians seem to forget that this is okay.

Also, meaning as function should be addressed. I don’t know of anyone who has interpreted this to a the language of mathematics. This definitely fascinates me. Definitions of terms in math are very rigorous. We tend to think of the meaning of terms as set in stone regardless of how they function in a proof or sentence. I think this is very much not the case as Lakoff and Nunez argue in their book. Sometimes we can use the same term in the same proof in two different ways if we are interpreting the metaphor in two different ways.

Just quick thoughts after rereading that wonderful work. I’ll post a follow-up tomorrow to see if I still feel that way.


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Penrose Objective Reduction

Moving on finally. I have recently come into contact with an “interpretation” of quantum mechanics that I was unfamiliar with (I think). I at least never looked it up in depth if I have heard of it. It is called Penrose Objective Reduction (or POR for short). It is very different than the standard theories in that it doesn’t ignore the collapse of the wavefunction (many worlds or decoherence). Instead it postulates something possibly more fantastic. Let’s start more towards the beginning before going there, though.

Take E=\frac{\hbar}{t} and interpret it. E is the degree of spacetime separation of the superpositioned particle and t is the time until POR occurs. This shows that small superpositions (i.e. things that are almost in a determined state) will take a long time to collapse objectively. This intuitively makes sense, since there isn’t really a “need” for it to take a determined position if it is undetermined at close to plank-length distance. It doesn’t contradict our world view. On the other hand, extremely large objects (say Schrodinger’s cat) will objectively collapse to a single position extremely quickly. This eliminates the “paradox” of Schrodinger’s cat.

But what exactly is POR? It is “objective” collapse of the wavefunction in that it doesn’t require an “observer” as other theories claim. Wavefunctions will naturally collapse. Will this collapse go to a random state (in which we know the probabilities, but still random)? Penrose says no. He claims that there is information embedded fundamentally in spacetime. He makes an even more extraordinary claim that it is “Platonic” in that it is pure mathematical truth, aesthetic, and ethical. Since I have spent weeks rejecting Platonistic views, I feel I should offer an alternative based on this method.

It is known that “empty” space (e.g. the mass gap, actually I can’t find it now, I was going to link it, I’ll keep looking) has enormous stored energy. Many people interpret this as where collective consciousness lives. Instead of some objective random collapse of the wavefunction, or some ethical godlike decision as to what it should collapse to, it seems as if we are missing the power of our own minds. Maybe a more karmic collapse. The thoughts and energy we put into the world gets stored in the area and influences the POR to give us back what we put out.

I didn’t get into a lot of aspects of POR, and I was just making up that last part on the spot, so it wasn’t very scientific or rigorous as to how it could work. Just some thoughts for today.


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The Philosophy of Embodied Mathematics

I’m really going to try to get off this topic soon. It is probably getting kind of old. What is the “romance of mathematics?” Lots of people talk about this (I think Hersh calls it the “front end”). It basically boils down to the view that math is objective and external to humans. If other intelligent life formed in a distant galaxy we would be able to communicate with them through math since it is universal. Math is the ultimate science, since it is absolutely precise and infallible. As such, math itself is the only science worthy to characterize the nature of math. And so on…

Embodied mathematics clearly rejects this viewpoint. Whenever it seems that we are studying objects external to us, or that math is appearing in nature (physics, fractals, etc) we are forgetting that this is being interpreted through our embodied mind first. We have no direct access to the external world. It only seems as if math is inherent in these places, since we have no other way of seeing it.

It is absurd to think that other life forms would develop the same mathematics as us. Their embodied mind might be completely different. They may have no conception of the metaphors we use. Mathematics has varied across cultures and history from different concepts in humans. Now we shouldn’t get too carried away. It is not purely culturally contingent. The similarities of math across cultures is perfectly explained from the embodied standpoint. Humans, regardless of culture, have many cognitive similarities. We embody certain mathematical concepts in the same way (e.g. we use image schema, conceptual blends, conceptual metaphors, etc).

There are no “absolute foundations” for math.

Mathematical truth is based on whether or not our embodied understanding of the subject is in accordance with the situation in question. Simpler: Truth is dependent on embodied human cognition.

Not only is Platonism rejected in the above sense, but formal reductionism is also rejected (these two being about 99% of working mathematicians’ viewpoints). Saying that all mathematics can be reduced to set theory and logic is false since that is ignoring the fact that mathematics is ideas. Formal reduction is a great metaphor with lots of use, but it is not all of math.

This is really only a brief philosophical analysis of this huge new discipline. I haven’t decided if I’m going to continue with another post tomorrow. If I do, then it will be on how this relates to my main interest (and it seems like the authors were semi-aware of this) of developing an aesthetic theory for mathematics.


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Last Book Club

Not having internet has been annoying since so much has been discussed in this book, and I wanted to get my ideas out before I forgot them. I tried jotting notes in the margins, but this post won’t do it justice. This will conclude the material of the book. There will still be a few more discussing implications that I’ve thought of while synthesizing the material. Here goes:

I’m still on the fence about the novelty of the idea of the BMI. This is the case even after many examples. It just seems that mathematicians already know they are doing this. Luckily, I am completely reverse of the reviews of this book. I think the first half was so-so, but it starts to kick out some great ideas in the latter half.

Chapter 9: Real Numbers and Limits. Some interesting commentary on why epsilon-delta definitions are difficult for students. Limits are conceptualized using the motion along a line metaphor for numbers, yet the definition is very static. It also accounts for ridiculous cases of epsilon (who cares about large epsilon? it is only the small epsilon that matter in limits). This is mathematically a necessity, but conceptually confusing. I agree. On to sums. Hmm…still not all that interesting. Just examples of the use of the BMI.

Chapter 10: Transfinite numbers. They talk about Cantor’s diagonal argument and some of the assumptions. The proof is usually taken to be formal, but in actuality it cannot be written down formally because you can’t express infinity as an actual entity. This means that the fact that there are more reals than rationals is inherently metaphorical. It is also discussed that Cantor’s one-to-one and onto definition for equivalent infinities is metaphorical and not absolute. It is one way to count infinities and see if they are the same. It is not the only way. We often lose sight of this, or not even realize it.

Note: Getting the picture yet? They are building this idea of metaphor to a pretty interesting climax.

Chapter 11: Infinitesimals. This was probably my favorite chapter because I had never seen a construction of the hyperreal numbers. They first builds what they call the granular numbers. This is essentially just the first “layer” of the hyperreals. You get the interesting result that I didn’t know about that there are number systems with no possible system of numerals (because it would need an infinite alphabet to express). This also brings up the concept that there are mathematical objects that are inherently metaphorical (since they can’t be expressed otherwise). I think that they think the most important part of this chapter is that “ignoring certain differences is absolutely vital to mathematics.” This refutes the idea of mathematics being perfect, exact, absolute, … , i.e. Platonistic. Yea! Finally, the big one comes out. This argument is much longer, but basically boils down to “calculus is defined by ignoring infinitely small differences.”

Chapter 12: Points and the Continuum. What to say about this… Basically it goes through the struggle of how to define a point. Do points on the real line touch? If they do, then by definition of having no length they are the same point. So they can’t touch. But the real line is continuous, i.e. there are no gaps. Thus points much touch. A paradox? Actually they break this down as a problem of blending two metaphors for talking about \mathbb{R}. This shows that when we talk about things as absolute truth, we may actually be referring to a metaphor which doesn’t exactly work in every situation. We must be careful what metaphor we are using and how it affects what we are talking about. Also the problem of attempting to discretize (write down mathematics in a precise and logical manner from axioms) the continuous is discussed. From a conceptual point of view this is impossible. In fact, it really hasn’t been insanely successful.

Chapter 13: Continuity for Numbers: The Triumph of Dedekind’s Metaphors. This talks about Dedekind cuts. Blah.

Chapter 14: Calculus Without Space or Motion: Weierstrass’s Metaphorical Masterpiece. This talks about how the geometric interpretation/metaphor for calculus had major limitations. There were functions that had nothing to do with motion. It talks about how Weierstrass extended calculus to work in these situations. Here again is the “choice of metaphor argument.”

Le Trou Normand. Here they give us the kicker. I’m going to do this in a more concise way. Construct the sequence of functions f_n(x)= n semicircles of perimeter \frac{\pi}{2^n} (where the first one starts at (0,0) and the last one ends at (1,0). Now each f_n has arclength \frac{\pi}{2}, but the sequence of functions converges to [0,1]. There is an apparent contradiction since the arclengths of the sequence converges to \frac{\pi}{2} and thus $[0,1]$ has “length” \frac{\pi}{2}.

The problem is the same as before. Our choice of metaphor is incorrect. We can’t say that the limit of the length of a pointwise convergent sequence of functions is the length of the limit under our current metaphor. But we can define a new metaphor in which this works. This is a common metaphor to use in functional analysis. Construct a function space in which our distance is d(f,g)=sup_x\big| f(x)-g(x)\big| +\int_0^1\left(\big|f'(x)-g'(x)\big|\right)dx. You can work this stuff out for yourself to see how it works.

Moral: Our choice of metaphor matters! Down with Platonism! We can’t treat functions as literally being curves in the plane or the motion of a particle. While these are useful metaphors at times, they should not be taken as literal objective representations that give us all the information and no excess incorrect information (careful on all the negatives I stuck in there).

Tomorrow: Philosophical implications.


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Book Club V

This chapter was about what the authors call BMI or Basic Metaphor of Infinity. Apparently most of the rest of the book is about examples of this. I don’t really get what is so revolutionary about it. Most mathematicians that I know already think about infinity in this way, so there is not much for me to write about. I actually found this chapter rather dull and tedious with all the examples.

BMI just says that we can embody infinity despite being finite by metaphorically considering infinity to be iterations of finite things without end. Oh, so looking through this chapter again right now, there are also lots of terms from cog sci and phil language that are defined, but also rather obvious. Like some terms naturally have endings (like jump inherently implies a landing) and some do not (like leaving does not have a termination, it is an open ended verb).

So there are two types of infinity, both subcases of BMI. The one that is conceptualized as something without and, and the one that is the “number” infinity conceptualized as a number greater than any other. Then there are examples…

I will be out of town for the next week and will not be able to update, but hopefully I’ll be done with the book by then and we’ll be able to move on to something a little more exciting.


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Math is not formalistic

Technically this is the next book club, but I think it deserves a title in its own right since I was planning on tackling it at some point. I have a friend that is a die hard formalist. He says that once we lay down the formal system of mathematics, everything is completely determined as true or false (ignoring obvious Godelian problems which he acknowledges). We don’t invent math precisely because we have to discover whether or not propositions are true in the formal system.

This argument is extremely difficult to refute. I can’t claim that we can invent a proposition that is false, but can be deduced through proof to be true (what he interprets as my “invention” standpoint). My argument essentially has been that mathematicians could care less about the truth of a proposition. You are missing the point of math if all you want to know is whether or not the Riemann Hypothesis is true. Mathematicians care about the ideas used in proof (invented things), and the method used (also invented). Thus mathematics is invented. Still, I had no hard way of convincing him. He still says that it is discovered since we couldn’t change the result no matter how it was done.

Finally! Lakoff and Nunez provide a good example as to why math cannot be considered purely formal. Basically chapter 7 is about our metaphorical constructions of sets. They basically talk about the fact that most confusion in math is the same term being used in two different ways (does this sound Wittgensteinian to you? All problems are actually just problems in language…). Philosophers call this the fallacy of equivocation. Now Lakoff and Nunez don’t really claim that this is an equivocation, instead they say that there is a definition and there are metaphorical ways to interpret that definition.

Lots of interesting examples in this chapter I wish I could write about, but here is the actual one. We can interpret an ordered pair (a,b) metaphorically in terms of sets as \{\{a\}, \{a,b\}\}. Check for yourself that this is a well-defined metaphor. Also, it is very common to interpret at a foundational level numbers as sets (look this up), but counting starting at zero would be \emptyset, \{\emptyset\}, \{\emptyset, \{\emptyset\}\}, etc. We confuse these metaphors to be actual definitions. This is not the case. We actually lose conceptually what we are talking about if we look at it that way.

E.g. What is the ordered pair of numbers (0,1) at this “foundational” level? Well, \{\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}. What is the set containing 1 and 2 as numbers (i.e. \{1,2\}). Well, exactly the same thing: \{\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}. These conceptual distinctions exist at a mathematical level (the ordered pair and the unordered set are not the same), but when we take the formalistic metaphor to be the definition we lose that distinction. All formalism is is one way of interpreting things. It is purely a metaphor. We lose concepts (as any metaphor by definition does not preserve everything). Thus, mathematics is not literally reducible to set theory.

Any counters to this? I believe this is my best weapon against formalists, and “discoverers” now.

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