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.

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.

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 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.

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.

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 . 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 semicircles of perimeter (where the first one starts at and the last one ends at . Now each has arclength , but the sequence of functions converges to [0,1]. There is an apparent contradiction since the arclengths of the sequence converges to and thus $[0,1]$ has “length” .

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 . 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.

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.

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 metaphorically in terms of sets as . 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 , 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 at this “foundational” level? Well, . What is the set containing 1 and 2 as numbers (i.e. ). Well, exactly the same thing: . 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.

Book Club IV

So. I didn’t get through as much reading as I thought I would. Two chapters isn’t bad, but nothing more than I would have done. There are a couple of interesting claims here. I’m still completely confused about how they are rejecting Platonism. I think that they are quasi-Platonists at best, maybe full out Platonists at worst. They beginning of chapter 5 talks about forms and essence. I kind of want to skip to the end and read their summary first just to get the big picture.

This is about the essence of algebra. They claim there is a general algebraic essence metaphor. The most interesting part of these two chapters is the claim that algebraic structure is not inherently in what we use like we naively think. The integers under addition does not have the group structure in it. We impose that metaphor onto the set in order to make more sense of it. This seems to make sense. It is just nothing I would have ever guessed or thought of on my own.

Next comes a partial answer to my Platonism question. I don’t feel like explaining the term folk theory (since I’ve used it so much in 20th Century Philosophy). We have a western folk theory of essence in mathematics. This is the reason we feel that we should try to characterize all of mathematics in the fewest number of independent axioms possible. We believe these axioms to describe the forms of mathematics. Then when we do math we are just discovering things about these essences. (Look up these terms as well if you are fuzzy on “form” and “essence”).

It turns out that the folk theory of essences does not describe mathematical cognition. When we talk about rotations of the triangle (cyclic group of order 3), the rotations and group structure are conceptualized independently. The thing that gets us from one to the other is the idea of “linking metaphors” (as opposed to the previously encountered grounding metaphors).

Hmm…I don’t remember coming across anything particularly interesting in chapter 6, which is about Boolean logic. It talks about the Classes as Containers metaphor as a basis for Boolean logic. This was already touched upon though. It just goes one step further to the symbolic-logic mapping that mathematicians use. This is how we get the feeling of “blind manipulation.” All of this traces back to the embodied container schema, though. The closing thought is that no matter how advanced our system of logic becomes, it will never be able to capture true human reason. Thus if we solely use logic for reasoning we will miss some things.

Commencement…again

So, six days after the last commencement I attended I now have to go give one of those speeches that I despise at my own commencement. Since all you lovely readers will not be able to make it to see me, I’ve decided to post my speech here (semi-philosophical in nature). Also, I will not have the time before I leave to read another chapter for a “book club” post. I will not be around until tomorrow night (but a six hour car trip that I am not driving should give me plenty of reading time).

Here it is:

I must admit that when I was asked to give this speech, my immediate reaction was, “Of course not!” I am very different from the majority of you in almost every aspect imaginable. I did not grow up in this area, I am not of the majority religion, my viewpoints on academics and learning are unlike the majority, and so on. This speech is supposed to be some sort of reflection on my years here that everyone here can relate to. I was the wrong choice.

Then I began to think about possible topics. Everything that came to mind was some blanket cliché that wouldn’t reflect the truth about my experience. After some more thought, I came to realize that rejecting this speech would just add one more layer to what I consider to be a four-year-long struggle with a single concept: specialization. Let me explain my situation:

I entered YSU as a music composition major. I became labeled at that moment. Every time I become labeled and forced into a box (quite literally in the practice rooms) I rebel. I can’t stand the idea of specialization. I then switched to a math major. Once I started getting good math, I became labeled again as a “smart math genius.” Like usual, I rebelled again. I started dabbling in physics and very seriously in philosophy as well. Finally, I wrote my honors thesis in the philosophy of aesthetics as a final effort to break my label of mathematician.

Why am I against specialization? This was a tough question. Somehow I realized that you miss something when you specialize. I saw this when examining some of the greatest achievements of mankind. From my own skewed viewpoint, one of these achievements is the work Gödel, Escher, Bach by Douglas Hofstadter. This great work examines consciousness from a wide variety of viewpoints including art, biology, computer science, mathematics, logic, and more. This truly original work of art presents some of the most comprehensive views on life I’ve ever read.

Douglas Hofstadter has a Ph.D. in physics. If he had let the world of specialization dictate his research and ideas, this would never have been possible. David Foster Wallace, an author, also talks about this idea of unification of all things. He was trained as a philosopher and mathematician, but writes novels now to express what he calls “the click.” Any work has “the click” if it somehow expresses life as a whole and not just some tiny detail. You may be wondering, “How does this affect you?”

We live in a world of forced specialization. There is no doubt about the effectiveness or necessity of specialization, but you must remember to not let it run your life. Whether you intend to enter the work force, become an artist, continue your education, or whatever, people will continue to force you into the box of their deciding. If you allow this to happen, you will probably become unhappy. This is because you are missing something: “the click.” Whenever you feel this happening, I urge you to take up something new. Look at life through different eyes. Hopefully, through time you will come to see the interconnectedness of all things.

This brings me back to my first point. I didn’t know how I was going to connect with you, because I was only focusing on the differences. I was focusing on the specialization. I had fallen into my own trap. I just needed to step back and realize that we are only on the surface different. Underneath it all, we are all connected. Thank you.

Book Club III

Chapter 4 has lots of repeats. It basically just gives more evidence that the four grounding metaphors are true plus some examples of them. There have been some really interesting new points brought up, though.

1. ERF’s (i.e. Equivalent Result Frames). This is a set of three things: a desired result, essential actions and entities, and a list of alternative ways of performing those actions with those entities to achieve the result. From the embodied mind standpoint, ERF’s are how we get associativity of arithmetic. Thus, according to this theory, associativity is not an axiom, but a consequence. I sort of like this. We’ll see how this stands up later (e.g. when we throw out associativity in non-associative algebras or loops).

2. The four grounding metaphors are isomorphic as mappings over the natural numbers. Well, I’d hope so. Also, the “proof” of isomorphism is a little sketchy. Also, as structures they aren’t isomorphic, since some of them produce things like fractions, irrationals, and negatives while others don’t. This isomorphism is only with respect to how they map to the naturals.

3. Uh. Why does Reuben Hersh endorse this book again? Numbers are things in the world, thus “Since things in the world have an existence independent of human minds, numbers have an existence independent of human minds. Hence, they are not creations of human beings and they and their properties can be ‘discovered.’” Well, I guess that settles the discovered vs invented question. Hersh is a hard-core invented proclaimer, though. Also, I’m not sure how much I buy the “things in the world have an existence independent of human minds.” I guess that depends on how you interpret the necessity of an observer in quantum mechanics.

Also, is this really what embodied mind is saying? Embodied mind is the idea that how we interact with these objects is how we perceive them and interpret them. Isn’t it true that if another species of intelligent beings formed somewhere, and they interacted with objects, in say a non-associative way, associativity would not be a consequence? In the same section we have, “There is only one true arithmetic, since things in the world have determinate properties.” I disagree precisely because of what I just said. There are many true arithmetics. In fact, in physics non-associative structures are rearing their ugly heads. If we had developed such that we saw and interacted with those structures before interacting with macro-structures, non-associativity could be the norm.

4. They are heavily pushing the metaphor idea. Good idea, but do all these metaphors they are referencing exist in all languages? This seems to be an English-centric argument. Maybe a certain person who has studied the cultural influence of mathematics can tell us if there are cultures that didn’t use, say, the motion metaphor. (i.e. 5 is close to 6 since we interpret motion from one natural number to the next as a short path).

5. Extension of subitizing to closure. I don’t remember if I mentioned this. The way we tell the difference between small numbers of things (up to about 4) is to subitize. We know whether something has 1,2,3, or 4 objects instantly. Groups of larger size get harder. We need to group into these smaller groups that we know to figure out how many are there. So this is not a closed operation. 2+3=5. We can subitize 2 and 3, but not necessarily 5. Why is it natural that we extend this to a closed operation? It seems unnatural actually, since the natural innate thing is not closed.

6. Probably the best thing about this chapter is the idea that (I can’t actually find it right now) symbols are important in the sense that the symbol representing the number (or whatever) is not the thing itself. A proper choice of notation is necessary to make calculating and manipulating as easy as possible. This has been my biggest soapbox argument for the past year. Let’s get rid of some of this outdated crap. String diagrams are amazing, box notation is amazing for algebra, categorification is amazingly compact, etc.

Now, don’t get me wrong. I love what these people are trying to do, but I think they aren’t even interpreting their own results properly. Maybe I’ll take my most cogent points from these posts and write a more well-founded and thought out essay to send to them.