Mathematics as a way of knowing

[Chinaski]

Here are two (very short) essays I wrote for a test (which I got an A- on ) on the role of mathematics as a way of knowing. I think it's a very interesting subject, and I hope to inspire a debate as well as provide interesting reading.


Please note that the use of the male pronoun to indicate humanity is not made without sufficient feminist awareness, it is merely a literary choice to aid with the flow of the text. In no way does the author intend to promote the idea that the male sex is more indicative of humanity and society.

Do we, as a society, value mathematical knowledge more than other areas of knowledge? Why? Which ones?

Although mathematics has existed for millennia among human cultures, and practical applications for it have existed ever since the Egyptian architects and pyramid builders, the reason why every single country in the world today has more hours of math in their public school curriculum than say, dance, music or even the humanities, is because mathematics has only found widespread use and usability among human society in later years, namely during the period of industrialization and the need for new languages that went with it. The industrial revolution completely changed the fulcrum of society, shifting the emphasis from the manual abilities of the agricultural farmer to the engineering abilities of the machine operator. This text does not seek to delineate the development of mechanical muscle or artificial intelligence; that is beyond the current scope of discussion. However, the obvious difference between manual, irregular, unpredictable agricultural labour and the rhythmic repetitions of the machine is that manual labour depends on the human undertaking it, and as such, is susceptible to all the ailments of the human, whereas machines are regular, predictable and unfailing (barring accidents or flukes of design, which only result from an error on the part of the human who designed them). Theoretically, machines (designed correctly) are perfect, because the mathematics involved in creating and operating them are relatively perfect. This is the reason for the current emphasis on mathematics in school curriculums. It has become necessary knowledge for understanding the machines we have integrated into the backbone of society. Out of pure survival must we now study mathematics rather than the humanities, yet is this truly a viable solution? It is a genuine and understandable concern, that of artificial intelligence, yet the true reason for this does not lie in the moral and ethical dilemmas that may emerge from too advanced intelligence. It lays in the dehumanization of society and the dependency on machines that function due to rigid, unfaltering, uncompromising thought systems.

Discuss the limitations of mathematics and why we need other forms of knowledge to fully understand and interpret the world.

Mathematics is only perfect when applied to its own field. It is not a technique that can be used to study the human being, his psyche or his emotions. This is because humans are such volatile beings, so susceptible to change and transformation that one cannot possibly hope to establish any sort of general rule regarding human behaviour. The argument changes, of course, upon entering the field of mob psychology, and sciences such as the fictional “psychohistory” (a term coined by scientist and author Isaac Asimov to describe the predictions of humanity based on a study of mass behaviour) seem to have true potential for practical applications, now more than ever. And yet, the disciplines that focus on the person rather than the machine, the arts and the humanities, are falling by the wayside as students spend countless hours struggling over equations and thought patterns that neither come naturally to humans nor will ever be practically applied by a human directly to his own life or existence.
Mathematics is not a human field of expertise. Even those among us who possess inordinate understanding of it are merely groping in the shadows of the twilight of a much darker night. Mathematics has found a practical application in machinery and computers, yet no human has ever run on matrices or binaries. Humans run on language, on subtleties, on hints and persuasions. To “operate” a person, it is necessary to trick him by means of rhetoric or mind games. Things like religion, political affiliation or nationalism are living tribute to this method. No person is easily programmable. And no person is without his exceptions and oddities that make it impossible to elaborate a general statement or formula to base his existence on. Mankind needs much more than that, and indeed, mankind is much better off dealing with more than that. Since mathematics exists outside of man’s creation, it is far too abstract a subject for him to fully grasp and understand, at least not at our current stage in intellectual and social evolution. What man can use with far more success is that which he creates himself. Literature will always be more human than math, as it changes with the societies that produce it. The same goes for music, dance, history, psychology and all the other humanities and arts. While it is true that mathematics can be applied to these subjects, it is not a full permeation, merely an attempt to adopt certain concepts and harness certain methods of thinking for a humanist purpose.

[Xamonas Chegwé]

Very well put, although I don't 100% agree with you on quite a few points - especially in the second piece.

I don't like your statement that "Mathematics is not a human field of expertise." While we are indeed "groping in the shadows" in some instances, this is true of any science. We can never "know it all", merely expand our horizons based upon what we already know. The world's top mathematicians have a profound understanding of their subject that the rest of us can only dream of. Physics, biology, chemistry, any subject you like, ALL lie outside of mans creation.

The same is true of any discipline - Literature included. While I can read a popular novel and glean pretty much as much from it as anyone, I would understand a 17th century Russian novel with far less clarity, having to rely on the skill of the translator, and guessing at certain historical, cultural and geographical nuances that would have been obvious to a contemporary audience. Add references to earlier literature, political satire and allegory, deliberate parodies of contemporary figures, passages on specialised subjects, double entendres, puns and other plays on words that only make sense in the original language, etc. and here I am groping in the shadows. Only a student of the history and language of that region and time would be able to truly appreciate the book.

What does set mathematics aside, in fact what makes it completely unique among all branches of human knowledge, is that its truths and formulae are not subject to the vagaries of the world in which we live. 1 + 1 = 2 in any universe. All of mathematics is built upon a very small number of axioms. From these, even the most complex formulae have been derived and rigorously proven. There is no universal truth outside of mathematics.

[Beelzebub2]

Well, I've been always interested in the connection of mathematics and music, how it can be applied and used in writing musical compositions, like J.S. Bach did, but many other compositors as well.

Scholars have spent a lot of time studying the way in which Bach made use of a number of formal mathematical patterns when he composed his majestic organ fugues. For instance, Bach used the "golden section" as well as the Fibonacci succession (1, 1, 2, 3, 5, 8, 13 etc., in which each number in the succession is the sum of the two previous ones). In many ways he worked like an architect, joining the two different parts of a musical piece into one harmonious whole before the actual process of composition started.

There were others as well, such as famous Renaissance composer Guillaume Dufay (c. 1400-1474), especially his polyphonic choral work, which he composed for the consecration of the cathedral in Florence in 1436. Some people have claimed that the form this motet takes, mirrors the architectonic number symbolism of the cathedral. They set out to prove that Dufay was portraying the proportions of the church in his music. Some interpretations of the work go as far as analyzing all notes and textual syllables in the motet so as to make the composition correspond to individual parts of the building.

Speaking generally, music is closely linked to absolute physical entities, such as frequency and the relation between intervals (an interval is the space between two notes). Already in Antiquity this was seen as the natural or cosmic premise on which music relied. Not just musical notation, but also the relationship between music and time has something to do with mathematics and with one of the most significant transformations in music history.

Polyphonic music recorded in notations, shows the length of each note in a uniform and measurable way, for the first time in Paris at around the year 1200. In order to indicate how the separate voices were to be coordinated in the work, composers had to make use of notations, which were also able to show the length of each note. This meant that they were further able to measure any temporal aspect, by dividing the length of each note into smaller units. This type of polyphonic music was not called "polyphony" as it is today, but "musica mensurabilis", or "measurable music". Musica mensurabilis opened new possibilities within musical forms which both Bach and Dufay developed further.

Mathematician and philosopher Boëthius (480-526) played an important part scientifically as well as in the understanding of music. Boëthius divided science into seven disciplines: grammar, dialectics, rhetoric, arithmetic, geometry, astronomy and music.

He viewed the first three disciplines as a single unit or whole, a "trivium", and the remaining four as another whole, which he named the "quadrivium". He based this categorization on the fact that the trivium had to do with language, whereas the quadrivium - which included music - had to do with numbers. The basis for Boëthius's world of ideas was the notion of music as audible numbers. He illustrated this with a legend about Pythagoras, the Greek mathematician and philosopher.

Based on the story about Pythagoras, Boëthius concluded that music is a matter of numbers. The medieval conceptualization of music also led to the view that music is a matter of numerical relations translated into sounds. Boëthius's and Pythagoras's approaches to music can be seen as ways of discovering already existing natural phenomena, phenomena which are all created by God.

In the beginning of the 14th century, a medieval theoretician of music wrote that 'music is about tones which are related to numbers and vice versa (about numbers which are related to tones)'. This notion is also found in the works of the philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716), who was the creator of analytic geometry. He wrote: "Music is the hidden arithmetical reckoning of the unconscious spirit".

For further reading...

[Chinaski]

@ XC: True, the knowledge you have of one subject depends on the context you find yourself in. The big difference with Mathematics is, as you said, that it is not relative. 2+2 will always equal 4. But isn't this true for physics and biology and all the other sciences? I don't agree on the literature, however. I think there is a big difference, and since literature is man-made, we are far more likely to understand it to a much fuller extent.


EDIT: Sorry, I'm not debating well at all. Put it down to being tired after a long day.

[Xamonas Chegwé]

@ XC: True, the knowledge you have of one subject depends on the context you find yourself in. The big difference with Mathematics is, as you said, that it is not relative. 2+2 will always equal 4. But isn't this true for physics and biology and all the other sciences? I don't agree on the literature, however. I think there is a big difference, and since literature is man-made, we are far more likely to understand it to a much fuller extent.


EDIT: Sorry, I'm not debating well at all. Put it down to being tired after a long day.

The same is not true of the other sciences. Newtonian physics was 'true' until Einstein blew it out of the water. All other science is subject to amendment in order to fit new evidence and discoveries. Maths however, is built upon an unchanging foundation - there will never be evidence that 1 + 1 = 3 (except in year 7's homework!) and so it never needs to adapt.

It is probably true that the physical laws that govern the universe are unchanging but there is no way of knowing exactly what these laws are in the same way as we can with Maths. We can refine our theories of physics and chemistry for a billion years and never know that we are right. With maths however, Pythagoras' theorem is as unshakably true now as it was when it was first discovered by the Chinese, Indians, Incas, whoever got there first.

On the subject of literature, there is much scholarly discussion about the meaning of works written within the last few decades and whole libraries have been filled with arguments about Shakespeare's plays. There are pieces of writing in languages that no-one speaks. Can you read THIS? Can you understand the text, let alone the meaning of that text? Do you speak Latin? Can you be sure that the meaning that we now attribute to Latin texts is the same as that that was intended? How about poetry? Do you understand Wordsworth, Goethe, Dante, as well as anyone?

Can you be sure that someone won't come up with a document that sheds new light on the meaning of a passage in some work of literature that was believed to be perfectly understood? Language is far more changeable than science - if the physical makeup of the universe can be reinterpreted, surely the text of a book is far more open to different readings?

I think you are confusing accessibility with understanding. Once you know the letters of the alphabet and have a sufficiently large vocabulary and a dictionary in a language, you can read any book in that language - sure - but that doesn't mean you can understand it in any depth. To do that requires a lot more time and effort reading similar literature - study - exactly the same kind of study required to understand science.

[Chinaski]

I think you are confusing accessibility with understanding. Once you know the letters of the alphabet and have a sufficiently large vocabulary and a dictionary in a language, you can read any book in that language - sure - but that doesn't mean you can understand it in any depth. To do that requires a lot more time and effort reading similar literature - study - exactly the same kind of study required to understand science.

Excellent point. I stand corrected.

[Xamonas Chegwé]

I think you are confusing accessibility with understanding. Once you know the letters of the alphabet and have a sufficiently large vocabulary and a dictionary in a language, you can read any book in that language - sure - but that doesn't mean you can understand it in any depth. To do that requires a lot more time and effort reading similar literature - study - exactly the same kind of study required to understand science.

Excellent point. I stand corrected.

I wasn't trying to correct you. Just debate the ideas. Don't be too quick to assume that I am right - I talk a lot of bollocks too! And my onions change with the weather.

[Chinaski]

I think you are confusing accessibility with understanding. Once you know the letters of the alphabet and have a sufficiently large vocabulary and a dictionary in a language, you can read any book in that language - sure - but that doesn't mean you can understand it in any depth. To do that requires a lot more time and effort reading similar literature - study - exactly the same kind of study required to understand science.

Excellent point. I stand corrected.

I wasn't trying to correct you. Just debate the ideas. Don't be too quick to assume that I am right - I talk a lot of bollocks too! And my onions change with the weather.

I know, I don't have the time or mental energy to address the other points right now, I was just pointing out that I did in fact confuse understanding with accessibility, something which takes away from my argument.

What did you think of the first topic?

[Xamonas Chegwé]

I think you are confusing accessibility with understanding. Once you know the letters of the alphabet and have a sufficiently large vocabulary and a dictionary in a language, you can read any book in that language - sure - but that doesn't mean you can understand it in any depth. To do that requires a lot more time and effort reading similar literature - study - exactly the same kind of study required to understand science.

Excellent point. I stand corrected.

I wasn't trying to correct you. Just debate the ideas. Don't be too quick to assume that I am right - I talk a lot of bollocks too! And my onions change with the weather.

I know, I don't have the time or mental energy to address the other points right now, I was just pointing out that I did in fact confuse understanding with accessibility, something which takes away from my argument.

What did you think of the first topic?

I'll let you know later. I am about to eat and go out.
(PS. If I don't post in a day or two, remind me!)

[Chinaski]

Well, I've been always interested in the connection of mathematics and music, how it can be applied and used in writing musical compositions, like J.S. Bach did, but many other compositors as well.

Scholars have spent a lot of time studying the way in which Bach made use of a number of formal mathematical patterns when he composed his majestic organ fugues. For instance, Bach used the "golden section" as well as the Fibonacci succession (1, 1, 2, 3, 5, 8, 13 etc., in which each number in the succession is the sum of the two previous ones). In many ways he worked like an architect, joining the two different parts of a musical piece into one harmonious whole before the actual process of composition started.

There were others as well, such as famous Renaissance composer Guillaume Dufay (c. 1400-1474), especially his polyphonic choral work, which he composed for the consecration of the cathedral in Florence in 1436. Some people have claimed that the form this motet takes, mirrors the architectonic number symbolism of the cathedral. They set out to prove that Dufay was portraying the proportions of the church in his music. Some interpretations of the work go as far as analyzing all notes and textual syllables in the motet so as to make the composition correspond to individual parts of the building.

Speaking generally, music is closely linked to absolute physical entities, such as frequency and the relation between intervals (an interval is the space between two notes). Already in Antiquity this was seen as the natural or cosmic premise on which music relied. Not just musical notation, but also the relationship between music and time has something to do with mathematics and with one of the most significant transformations in music history.

Polyphonic music recorded in notations, shows the length of each note in a uniform and measurable way, for the first time in Paris at around the year 1200. In order to indicate how the separate voices were to be coordinated in the work, composers had to make use of notations, which were also able to show the length of each note. This meant that they were further able to measure any temporal aspect, by dividing the length of each note into smaller units. This type of polyphonic music was not called "polyphony" as it is today, but "musica mensurabilis", or "measurable music". Musica mensurabilis opened new possibilities within musical forms which both Bach and Dufay developed further.

Mathematician and philosopher Boëthius (480-526) played an important part scientifically as well as in the understanding of music. Boëthius divided science into seven disciplines: grammar, dialectics, rhetoric, arithmetic, geometry, astronomy and music.

He viewed the first three disciplines as a single unit or whole, a "trivium", and the remaining four as another whole, which he named the "quadrivium". He based this categorization on the fact that the trivium had to do with language, whereas the quadrivium - which included music - had to do with numbers. The basis for Boëthius's world of ideas was the notion of music as audible numbers. He illustrated this with a legend about Pythagoras, the Greek mathematician and philosopher.

Based on the story about Pythagoras, Boëthius concluded that music is a matter of numbers. The medieval conceptualization of music also led to the view that music is a matter of numerical relations translated into sounds. Boëthius's and Pythagoras's approaches to music can be seen as ways of discovering already existing natural phenomena, phenomena which are all created by God.

In the beginning of the 14th century, a medieval theoretician of music wrote that 'music is about tones which are related to numbers and vice versa (about numbers which are related to tones)'. This notion is also found in the works of the philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716), who was the creator of analytic geometry. He wrote: "Music is the hidden arithmetical reckoning of the unconscious spirit".

For further reading...

I've never been able to grasp the mathematical workings of music. Even bands like Meshuggah drive me insane. It's very interesting however, do you study music?

[Xamonas Chegwé]

Do we, as a society, value mathematical knowledge more than other areas of knowledge? Why? Which ones?

Although mathematics has existed for millennia among human cultures, and practical applications for it have existed ever since the Egyptian architects and pyramid builders, the reason why every single country in the world today has more hours of math in their public school curriculum than say, dance, music or even the humanities, is because mathematics has only found widespread use and usability among human society in later years, namely during the period of industrialization and the need for new languages that went with it. The industrial revolution completely changed the fulcrum of society, shifting the emphasis from the manual abilities of the agricultural farmer to the engineering abilities of the machine operator. This text does not seek to delineate the development of mechanical muscle or artificial intelligence; that is beyond the current scope of discussion. However, the obvious difference between manual, irregular, unpredictable agricultural labour and the rhythmic repetitions of the machine is that manual labour depends on the human undertaking it, and as such, is susceptible to all the ailments of the human, whereas machines are regular, predictable and unfailing (barring accidents or flukes of design, which only result from an error on the part of the human who designed them). Theoretically, machines (designed correctly) are perfect, because the mathematics involved in creating and operating them are relatively perfect. This is the reason for the current emphasis on mathematics in school curriculums. It has become necessary knowledge for understanding the machines we have integrated into the backbone of society. Out of pure survival must we now study mathematics rather than the humanities, yet is this truly a viable solution? It is a genuine and understandable concern, that of artificial intelligence, yet the true reason for this does not lie in the moral and ethical dilemmas that may emerge from too advanced intelligence. It lays in the dehumanization of society and the dependency on machines that function due to rigid, unfaltering, uncompromising thought systems.

I am not sure that Egyptian architects were the first to apply maths. They may be the earliest record we have of such a thing (although I'm not even sure about that) but I would be surprised if maths wasn't being utilised pretty much as soon as it was invented - in fact, I would go as far as saying that it would not have existed at all if it didn't have a use. The complexity of Egyptian geometry and their understanding of fractions is well documented, but far earlier records of mathematics have been discovered. Quantifying time, currency, enumerating items for barter - all of these are practical applications of maths that long predate the Pharaohs.

I would broadly agree that the rise of technological society has necessitated an increased understanding of maths - or more accurately, an increased understanding of increasingly complex maths. Although it has often been the case that the maths has long predated the technology - eg. complex numbers were seen as an interesting philosophical diversion until practical applications were discovered in electronics (most noticeably in signal analysis.)

I think that the greatest reason for the preeminence of maths in modern education is not so much an increase in its practical uses, as these have generally followed from scientific discoveries that have followed in turn from the maths, but rather a move away from a metaphysical view of the world towards a more rationalist, scientifically focussed one. It wasn't humanities that earlier scholars concentrated on, it was theology and (religiously focussed) philosophy. Look at the works of Descartes, Pascal, Newton - for every scientific work that these great thinkers produced, they produced a whole pile of philosophical musings about the nature of god. It is important to remember that subject specialisation is a recent development; there were no mathematicians, physicists, biologists, geologists, philosophers, historians in universities a few hundred years ago, learning was far more generalised. In fact, many universities grew from cathedral schools or monasteries. Again, it was the shift in thinking (and power) from the religious to the temporal that led to subject specialisation.

Anyway, that's an alternate view on how maths grew in prominence.

[Beelzebub2]

Well, I've been always interested in the connection of mathematics and music, how it can be applied and used in writing musical compositions, like J.S. Bach did, but many other compositors as well.

Scholars have spent a lot of time studying the way in which Bach made use of a number of formal mathematical patterns when he composed his majestic organ fugues. For instance, Bach used the "golden section" as well as the Fibonacci succession (1, 1, 2, 3, 5, 8, 13 etc., in which each number in the succession is the sum of the two previous ones). In many ways he worked like an architect, joining the two different parts of a musical piece into one harmonious whole before the actual process of composition started.

There were others as well, such as famous Renaissance composer Guillaume Dufay (c. 1400-1474), especially his polyphonic choral work, which he composed for the consecration of the cathedral in Florence in 1436. Some people have claimed that the form this motet takes, mirrors the architectonic number symbolism of the cathedral. They set out to prove that Dufay was portraying the proportions of the church in his music. Some interpretations of the work go as far as analyzing all notes and textual syllables in the motet so as to make the composition correspond to individual parts of the building.

Speaking generally, music is closely linked to absolute physical entities, such as frequency and the relation between intervals (an interval is the space between two notes). Already in Antiquity this was seen as the natural or cosmic premise on which music relied. Not just musical notation, but also the relationship between music and time has something to do with mathematics and with one of the most significant transformations in music history.

Polyphonic music recorded in notations, shows the length of each note in a uniform and measurable way, for the first time in Paris at around the year 1200. In order to indicate how the separate voices were to be coordinated in the work, composers had to make use of notations, which were also able to show the length of each note. This meant that they were further able to measure any temporal aspect, by dividing the length of each note into smaller units. This type of polyphonic music was not called "polyphony" as it is today, but "musica mensurabilis", or "measurable music". Musica mensurabilis opened new possibilities within musical forms which both Bach and Dufay developed further.

Mathematician and philosopher Boëthius (480-526) played an important part scientifically as well as in the understanding of music. Boëthius divided science into seven disciplines: grammar, dialectics, rhetoric, arithmetic, geometry, astronomy and music.

He viewed the first three disciplines as a single unit or whole, a "trivium", and the remaining four as another whole, which he named the "quadrivium". He based this categorization on the fact that the trivium had to do with language, whereas the quadrivium - which included music - had to do with numbers. The basis for Boëthius's world of ideas was the notion of music as audible numbers. He illustrated this with a legend about Pythagoras, the Greek mathematician and philosopher.

Based on the story about Pythagoras, Boëthius concluded that music is a matter of numbers. The medieval conceptualization of music also led to the view that music is a matter of numerical relations translated into sounds. Boëthius's and Pythagoras's approaches to music can be seen as ways of discovering already existing natural phenomena, phenomena which are all created by God.

In the beginning of the 14th century, a medieval theoretician of music wrote that 'music is about tones which are related to numbers and vice versa (about numbers which are related to tones)'. This notion is also found in the works of the philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716), who was the creator of analytic geometry. He wrote: "Music is the hidden arithmetical reckoning of the unconscious spirit".

For further reading...

I've never been able to grasp the mathematical workings of music. Even bands like Meshuggah drive me insane. It's very interesting however, do you study music?

Yes, I play the oboe, thanks for asking.
There is a possibility that in two years from now I enroll the Academy.

[Chinaski]

Yes, I play the oboe, thanks for asking.
There is a possibility that in two years from now I enroll the Academy.

Awesome, my Dad has a masters in philosophy of music and works at the Conservatory here, I've constantly got discussions floating around and I pick it up. It's extremely interesting, but I find it a bit too convoluted.

[Brian Peacock]

[...] Theoretically, machines (designed correctly) are perfect, because the mathematics involved in creating and operating them are relatively perfect...

I feel you need some theory of perfection in order to support this article. Is not the mathematics involved in a game of football also perfect? Is not imperfect mathematics just wrong mathematics or incorrectly conceived and/or processed mathematics? How perfect is relatively perfect? As you suggest, a poorly conceived machine is 'perfect' if it matched the designers conceptions and expectations, and regardless of its ability to perform or fulfil the task it was designed for efficiently - or perfectly. This is not within the sphere of mathematics.

[Xamonas Chegwé]

[...] Theoretically, machines (designed correctly) are perfect, because the mathematics involved in creating and operating them are relatively perfect...

I feel you need some theory of perfection in order to support this article. Is not the mathematics involved in a game of football also perfect? Is not imperfect mathematics just wrong mathematics or incorrectly conceived and/or processed mathematics? How perfect is relatively perfect? As you suggest, a poorly conceived machine is 'perfect' if it matched the designers conceptions and expectations, and regardless of its ability to perform or fulfil the task it was designed for efficiently - or perfectly. This is not within the sphere of mathematics.

FS,

Show me a perfect machine? One that neither wears out, requires servicing nor needs lubricating? It doesn't exist. And it wouldn't exist even if the maths behind it was absolutely perfect, let alone relatively perfect - whatever that means! - Perfect/not perfect, no exists relatively [/yoda]

Entropy and the unpredictable nature of reality have a way of fucking up any machine. That doesn't make the maths wrong.