Mathematics: Invented or Discovered?

For my history of math course we were asked to write an essay on the topic "Is math invented or discovered?" This is my response.

The debate regarding the ontology of mathematics is a philosophical quandary that extends deep into our cognitive history, near the emergence of sincere cognizance itself. This fact is hinted at in the division of arguments, in which a significant subset is Platonism. The classic allegory of the cave is often illustrated with a specific chair in the room as an oblique projection of the form that unifies all chairs under the notion of chair-ness, some vague set of qualities that result in an object being classified as a chair. This approach does elicit some glimmer of understanding, but the allegory has a vastly more ornate interpretation with consideration of the forms as abstract mathematics and the shadows as specific instances of those general principles. From this perspective there is some credence in the conjecture that Plato was influenced by the ontology of the Pythagoreans, which held that “Everything is number.” With mathematical forms as eternal and unchanging, a Platonist concludes that mathematics is discovered. 

 

A common reaction to this conclusion is the proposed problem of a priori existence, which cites the contradiction that the nonphysical forms exist without a physical manifestation before attaining representation as chemicals in the brain. If mathematics can only exist as an arrangement of physical objects, and these arrangements can only be produced by consciousness, it is reasonable to conclude that mathematics is invented through intelligent processing of experience. 

 

Despite so much thought on the invention vs. discovery of mathematics, the question is broken--clearly a false dichotomy--which really should have been recognized after all the contradictions started arising. It might be helpful to approach this question with a set theoretic interpretation of language. Let each word be a set comprised of its synonyms, including itself (as a singular element), and its definitions. Considering the word roots A = “invent” and B = “discover,” most modern references will give A intersect B as not null, frequently even giving the subset {invent, discover}. Now the question is if, for the word “mathematics” = M, 

 

(A is a member of M) OR (B is a member of M)

but this is a misrepresentation of the problem since A and B are not mutually exclusive, thus mathematics might be a member of invented, discovered, both, or neither.

Both the failure and success of language are a result of its persistent nebulosity, which enables the vaguely logical cogitation that profoundly influences our consciousness; to this we owe our capacity to experience the wonder of metaphor and the sincere difficulty of attaining certainty. As experienced language users, we know that what constitutes a word is not limited only to definitions and synonyms, that language does not naturally obey logic, which is why non formalized philosophical debate can proceed indefinitely. Accordingly, we can redefine the question and approach from another direction entirely, with definitions that appear mutually exclusive. Consider the two statements that thefreedictionary offers on the page for “discover”:

• We discover something that existed but was not yet known.
  
• We invent something that was not in existence.
  

Note that the second statement implies something, for if something was not in existence, it must not have been known, so

• We discover something that existed but was not yet known.
  
• We invent something that was not in existence and was not yet known.
  

which reduces to

• We discover something that existed.
  
• We invent something that was not in existence.
  

Thus distinguishing between invention and discovery relies entirely on existence; now we must ascertain if there is a difference between existence and non-existence. Consider the following definitions taken from thefreedictionary:

• Exist: To have actual being; be real.
• Existence: The fact or state of existing; being.
• Being: The state or quality of having existence.
• Real: being or occurring in fact or actuality; having verifiable existence.
• Actuality: The state or fact of being actual; reality. See Synonyms at existence.
• Actual: Existing and not merely potential or possible. See Synonyms at real.
• Fact: Something demonstrated to exist or known to have existed; believed to be true or real.
• True: Consistent with fact or reality; not false or erroneous. See Synonyms at real.

An attempt to simplify the definition of “exist” by replacing words with their definitions results in nonsense along the lines of

• Exist: To have the fact of existence; having existence in fact or the fact of having existence; having existence or occurring in fact or the fact of having existence existing; having verifiable existence.

In lieu of a definition that consists of something other than self substantiation, let the definition of existence be the following,

• Exist/Existence: true.

where “true” is in accordance with the familiar logic construct. Now we have a definition which is very useful for a formal analysis of the problem. If non-existence is not true, then there is no non-existence and everything exists; if non-existence is true, then non-existence must actually be existence by definition, therefore everything exists and everything, including mathematics, is discovered.

But this answer is contrived and not inviolable, because the resolution of the problem in the system of logical analysis, like all, depends entirely on the definitions. We can reach the opposite conclusion by giving an alternate definition, which would clearly result in mathematics classified as invented:

• Existence: the quality gained by something when it is first represented in a human brain.

Thus, in order to answer a question, the terms must be well defined, which is not the case for this ontological debate.

CS Senior Project Proposal: Earfingers

Within this milieu of stupendously sophisticated technology and virtual reality, humanity has indubitably extended the boundary of what is possible effectively beyond limits set by imaginations all but exceptional. The aim of this project is to tax this overabundance in order to produce a novel tactile human/computer interface with the potential to enable an unprecedented interaction with information, and a chance at evoking a new experience of existence through the manipulation of sensation mechanisms; that is, to possibly feel something that has never been felt before.

There is no question regarding the significance and success of the human brain, exemplified by artifacts such as smartphones that are capable of performing feats essentially indistinguishable from magic. It is remarkable we are able to interpret and utilize information so effectively when the process depends entirely on just five low-resolution information pathways better known as senses. Recognizing the inherent limitations in these senses and manipulating our environment in effort to circumvent them has continually and directly expanded our ability to understand existence---for instance the development of the optical microscope immediately led to the incredibly profound revelation that all life is composed of legions of fundamentally similar cells. Despite over three hundred years of this knowledge in the public domain it remains practically inconceivable, as what we think we know more than anything else, of ourselves and of others, is based only on something vaguely real, each of us being emergent properties of a cloud of many Trillions of individual cells instead of the one continuous thing we imagine. That conscientious enhancement of sensation was necessary, and that we are able to operate oblivious to our true nature as an Astropolis of individuals indicate that our behavior is at least informed by sensation; had the ability to distinguish and accurately record individual cellular behavior been a sixth member of our sensory repertoire, it is certain that biology would be extraordinarily more advanced than at present, and it is likewise certain that both the way we experienced existence and the way we interacted as a society would be fundamentally different. For another example imagine that we had the ability to sense blood flow in another brain (like functional magnetic resonance imaging): instead of having once learned the abstract fact that localized hemodynamic response can indicate something relatively specific like ongoing hunger/hunting drive, we could use our sense and proven intellect to determine that what is apparently a log at the watering hole according to five senses is actually a hungry crocodile with information from the sixth. Clearly then sensation doesn't just inform behavior, but in many important ways defines boundaries for it by serving to enumerate possible outcomes. Unfortunately my project is not to create a portable, affordable FMRI; instead we have gone so far afield to illustrate just how significant sensory experience is. That we have but five information pathways is a fact not readily malleable (until we start mechanically re-engineering the nervous system), and this is a major disappointment in light of the significance of sensation and the potential of supernatural sensation to redefine our experience. The objective of this project is essentially to experiment with several softhacks (no rewiring necessary) on the nervous system based on what is known about how it operates. Instead of trying to stimulate novel neurological information and make it perceptibly informative, I will try to feed known neurological information down pathways intended for other information in hope that the signal will nonetheless be at least partially recognized. In particular, I will be attempting to induce the perception of sound through the tactile and visual senses.

The majority of sound is perceived through a single information pathway into the cerebrum; this in itself is not surprising, but that doesn't necessarily mean this information can't enter through another pathway. It's a well known result of neuroplasticity that any region of the brain not receiving the intended sensory input may be used for the processing of other senses, a phenomenon observed in the utilization of the occipital lobe in the congenitally blind. There's evidence that tactile sensation of vibration excites activity in the auditory cortex of the normal human encephalon. There's another neurological relationship between auditory and tactile information in the functional limitation of operational frequency, in other words, the neurons can only convey an action potential (assumed to operate ultimately as a decision problem) from approximately 20-1000 Hz due to the time it takes to pump ions across the cell membrane to recharge. The optimum response frequency, where the action potential is in synch with the stimulation source, of the Pacinian and Meissner mechanoreceptors corresponds with the frequency at which the large outer surface of the cochlea responds in synchrony---both from about 50 to 200 Hz. The cochlea responds to stimuli up to around 20 KHz by taking advantage of signal aliasing and the tendency for higher frequencies to travel deeper into its coiled structure to discretize the soundwave into what is called a tonotopic mapping. Since the cochlea has about 3600 total receptors, the signal being sent to the brain is more or less a 6 ms frame length frequency decomposition into 3600 dimensions with an indicator function if ignoring amplitude. Theoretically, then, reproducing the neurological signal precisely would require as many tactors. The specific spatial distribution of mechanoreceptors remains to be found, otherwise it's generally figured as more on the fingertips than on the torso.

For my project I intend to create the experimental hardware and software in effort to reproduce sound information through the tactile and visual pathways. The tactile information will be generated by moving magnet voice coil tactors (a portmanteau of tactile and motor), driven by an Arduino receiving FFT results from a PC. I intend to have a minimum of 10 tactors, which is more relevant than the upper boundary of 3600 per hand. Scaling the number of tactors will necessitate efficiently encoding the addresses and data over the serial bus so the Arduino can stay in synch, which can be avoided as the 16 MHz clock of the ATmega328 could signal quite a few receptors at 200 Hz. Then there's the delectable sundry mechanical quandaries, which will need be overcome one way or another. For visual stimulation, I will utilize the common, powerful display technology and seek to create a program that presents a neurologically meaningful visual representation of sound, trying to account for the idiosyncrasies of the visual system such as the relatively small area of high resolution on the retina, saccades, and distribution of rods/cones. Whether this concert of stimulation will have any effect remains to be seen, but even no effect on those born with normal audition nonetheless leaves the possibility of eliciting the perception of sound for the congenitally deaf, and even just the chance of enabling someone to experience music for the first time, the possibility of being able to share the uniquely human celebration that is music is justification enough for my efforts. Because this project is experimental in nature, and, excitingly, the results unknown, I think the single best indication of success will be that the people who evaluate what I have done are impressed by it.

Blonde Redhead - Falling Man

Blonde Redhead has been having a sincere impact on my life by provoking such a profound sound for somewhere near six years now. Despite so much time, and only because I had intended to post this, I just noticed that the person on the cover is partially uncovered. I'm certain the image was intentionally constructed so the intensity of the face distracted the much subtler salaciousness; I suppose then I must admit that this remarkable band isn't just expanding my mind with sound, but with images as well. The combination of auditory and visual information is apt, as my senior project is an attempt to enable the combination of those two with touch in addition. So far I believe it has been progressing swimmingly, which is good, because my advisers have already expressed their concern that this project is too large. It is almost certain that the results will be blogged, so stay tuned.