MTH 141/161: Previous Tests

Previous Tests

Unit #5: Equations (Study Guide)

Study Guide 5

Unit #4: Introduction to Algebra (Study Guide #4)

Study Guide 4

MTH 161: Test #1 Solutions

MTH 161 Test 1 Solutions

MTH 141: Test #1 Solutions

MTH 141 Test 1 Solutions

Representing Nature

Our readings this week follow nicely from our discussion last week on order and representation. From our readings on objectivity, it is clear that our notions of objectivity have changed over time, and that our attempts to place the mirror of science up against nature has always been mediated/contaminated by our technologies/culture.  This led me to think: have we ever been able to paint a clear picture of what is   there in nature, or is there more than meets the eye? 

This week’s readings got me thinking about molecular renderings in the health sciences.  Today, molecular visions are providing scientists with key pieces to the great puzzle of disease prevention.  X-ray crystallography is a technique that renders 3-D molecular structures of viruses/proteins, and once these structures are uploaded onto the protein data bank, mathematicians can study these structures to gain a “feel” for how these viruses bend and twist in space.  Locating a virus’ weak structural points can unlock key information about how drugs can be designed to bind and penetrate through these structural weak points.   What I would like to focus on here is: how we can even come up with a structural image of a virus in the first place?  But, firstly, let’s talk some more about mathematics.     

What is important to consider in this discussion is the role of mathematics in assisting the technology to render 3-D structures of proteins.  As I mentioned earlier, all of our renderings contain traces of our culture.  Mathematics, as we know it today,  is a symbolic technology that allows us to organize numerical data about the world.  Mathematicians can then find patterns in the data to understand, for example, the rhythmic patterns of the heart.  In this particular case, mathematical algorithms (drawing from probability theory)  are written into x-ray crystallographic technologies to render molecular structures of proteins.  The mathematical signature is what renders the probable locations of the electrons.  This information, in turn, provides us with density cloud configurations for each electron.  Connecting this technique to Barad’s work on agential realism, if  scientific phenomena (e.g. molecules) is constitutive of reality, then the molecular model, according to Barad, would be a rendering  of the “thing” inside of the phenomena itself, not the “thing” behind the phenomena [studies in the sociology of mathematics reveal that the belief in eternal non-physical Platonic forms - "things" that are out there in nature waiting to be discovered - continue to pervade throughout our educational institutions (Brown, 2009)].  But here comes the twist.  Reality, for Barad,  is not composed of things-in-themselves, or things behind the phenomena.  These “things” are not isolated entities awaiting their discovery.  Rather, we play an active role in creating imaginings of the world (both human and non-human actors are involved in this process).  They become real because we are bound to these “things” of the world through our performativity with them.  We are not passive observers of an external reality.  The  things we find (or co-create) inside of the phenomena come into being as we intra-act with nature.  What an amazing idea! But I wonder if this only applies to realities on the atomic and subatomic levels. 

Again, X-ray crystallography is not unmediated, nor is it uncontaminated by culture – it surely does not provide us with an exact image of what is out there in nature.   However, these images are ultimately being used to design pharmaceutical drugs to fight off disease.  I find this fascinating! 

Agential realism has as an appeal to it.  There is this feeling of spontaneity/randomness in the simultaneous process of understanding both the world and ourselves, as was the case in Bohr’s atomic research.  What is interesting for me is the field of possibilities that can emerge from these intra-actions.  This led me to think: why has our culture placed so much emphasis on molecular biology, an approach that probes the smallest levels of life,  to solve health related issues?  If there are other possibilities for imagining and treating disease (e.g. Qigong , Acupuncture, Moxibustion, etc.), then how does one approach overarches the others?  In Situated knowledges, Donna Haraway discusses the importance of considering multiple subjectivities, including an embodied understanding of the world.  I believe that her initiatives have been taken up by Barad in a rather interesting way.

[Sources]

Barad, Karen. (2003). Posthumanist Performativity: Toward an Understanding of How Matter Come to Matter, Signs 28/3:801-31.

Bishop, Alan. (1988). Mathematical Enculturation (Kluwer Academic Publishers, London).

Brown, Richard. (2009). Are Science and Mathematics Socially Constructed? A Mathematician Encounters Postmodern Interpretations of Science (World Scientific). pp. 163-164

Sensing Aliveness

Daniel Heller-Roazen’s book The Inner Touch challenges our modern day notions of what constitutes self-consciousness (its classical Greek origin: sunaisthesis, which contains no reference to a “self”), including where we locate this inside the human body.  Today, some would say that consciousness resides in the brain, and this is reflective of a long tradition of philosophical/scientific inquiry into the very essence of our being.  It is representative of a particular thought that style that has dominated our imaginings/renderings of the self.  Roazen, on the other hand, invites us to re-consider the age old problem; of our ability to sense the world, and, most importatly, our ability to monitor ourselves in the process.  The question arises: to what can we attribute the sense that we are sensing?  Roazen provides a detailed sketch of how this question was tackled in different historical periods.  In painting this picture, Roazen begins by taking a closer look at Aristotle’s writings on sensation.  Roazen makes it very clear at the outset of his book that “…the ancients spoke little of consciousness and a great deal of sensing” (p.21).  This is the starting point from which we are understand Roazen’s work, that is, we cannot be too quick to draw any sort of connection between Aristotle’s notion of the common sense to our modern notion of consciousness.  Roazen suggests, on the other hand, that this “common sense”, “master sense”, or “inner touch”, can be correlated to the sense of touch, which is, according to Aristotle, “the most acute of all man’s senses” (p.292).  Roazen goes on to say “…in the ancient doctrine the power to think finds its roots in the tactile faculty and nowhere else” (p.294).  This perspective invites us to think about how self-awareness,  or how our modern notions of consciousness might rest on this simple idea.  Roazen, near the end of the book, poses the important question “What would it mean for touch to be the root of thinking, and for thinking, in turn, to be its most elevated form a kind of touch?” (p.295).  What if our ability to touch precludes our ability to sense our own aliveness?  This challenges the Cartesian principle “I think, therefore I am” as Aristotle’s principle, which links existence to sensation, would read ”I sense, therefore I am” (p.61). There is good reason to believe that Aristotle’s stance on sensing aliveness stems from his conviction that “…man becomes intelligible by touching” (p.295).  Our very own existence, including our sense of our own well being and connection to the outside world, however fragile, might rest on this notion.  In the case where Roazen documents, in the chapter entitled Phantoms, George Dedlow’s consciousness of sensing that that which he has not (in this case, his lost limb), he draws a very clear connection between our sense for our own aliveness and our ability to sense that we can touch, “Having lost part of his body, he imagines he will be forever sentenced to a sentiment of corresponding loss in his “own existence”" (p.258).  George Dedlow’s loss in his ability to feel his own aliveness destroys his relation to the outer world – the very world that affirms that he is alive – and ultimately makes him feel that he is slowly degenerating inside.  This brings us back to the central question: if our aliveness (no to mention our well being) is so heavily weighted on our ability to touch, then what does this say about our modern notions of self-consciousness? Is self awareness, or consciousness, a cognitive process or is it a process that we would attribute to sensation? What if self consciousness, which we would normally attribute to the cognitive process, derives from our ability to touch and feel the world around us? 

What I found most interesting was Roazen’s chapters on Sleep and Awakening.  Aristotle states in his De Somno that sensation is ” …a kind of movement of the soul by the intermediary of the body” (p.66).  It is said that Aristotle thought a great deal about the sensing power of  dormant animals.  On the nature of dreams, Aristotle claims that “…it is not by sensation that we sense the dream” (p.70).  However, Aristotle goes on to say “In a certain way, the dream is a sensation” (p.70).  Although these are contradictory claims, Aristotle believes that there is a “further dimension of sensation” that in “both the presence and the absense of waking…there is something in the soul that does not cease to sense” (p.71).  This “something” , which resides in the soul, is, for Aristotle, the voice that says “the apearance is a dream” in order to awaken the animal (p.71).  But Aristotle gives no reason as to why there might be “something in the soul that says that the appearance is a dream.”  For all we know, the dream might feel as real as the waking state!  This reminds me of a story by the Chinese philosopher Chuang Tzu (4 century BCE).  The story is called 莊周夢蝶 Zhuāng Zhōu mèng dié (Chuang Tzu dreamed he was a butterfly).  It reads:

“昔者莊周夢為蝴蝶，栩栩然蝴蝶也，自喻適志與，不知周也。俄然覺，則蘧蘧然周也。不知周之夢為蝴蝶與，蝴蝶之夢為周與？週與蝴蝶則必有分矣。此之謂物化。”

Translation:

Once Zhuangzi dreamt he was a butterfly, a butterfly flitting and fluttering around, happy with himself and doing as he pleased. He didn’t know he was Zhuangzi. Suddenly he woke up and there he was, solid and unmistakable Zhuangzi. But he didn’t know if he was Zhuangzi who had dreamt he was a butterfly, or a butterfly dreaming he was Zhuangzi. Between Zhuangzi and a butterfly there must be some distinction! This is called the Transformation of Things.

(2, tr. Burton Watson 1968:49)

This story blurs the lines between sleeping and waking, but at the same time it is a lesson on awakening.  Chuang Tzu asks: Where is reality? Is it in my dream or is it in my waking state? Am I the butterfly’s dream, or is the butterfly my dream? Where am I? This is an important question relating to our inner awakening. Roazen begins the chapter Awakening with the powerful statement “sleep ends in waking” (p.73).  But how do we know that we are not dreaming?  Sensations in the dream world seem as real as it would in our waking state, so how can we really know that our so-called waking state is not itself a dream? Where is reality and how can we become more aware of our aliveness?  These are the central questions in Chunag Tzu’s story.  Paul Valery says “there is no more stimulating phenomena for me than awakening” (p.75).  Valery goes as far as to say that ”the self comes to be in awakening” (p.76).  This has a very spiritual dimension for me.  And this is why I chose the story of the butterfly.  I believe that the story is a rendering of what it means to be alive and what it means to sense something that is real.