Folktale Structure as the Key to the Success of the Harry Potter Series

This is an html-friendly version of my research project on the Proppian folktale structure of the Harry Potter series. The pdf may be downloaded here. This was first presented at “A Brand of Fictional Magic” conference at St Andrews University, Scotland, in May 2012 (pdf of the program here), and updated for the Southwest Popular/American Culture Association meeting in February 2013. A final version will appear in print ca. Fall 2015.

The Question

“Why has the Harry Potter series of books been so popular?”

Hypothesis

The popularity of the Harry Potter series is due to the books’ narrative structure, in particular, its concordance with a linear sequence of elements typical of folktales1 as outlined by Vladimir Propp. The aesthetic satisfaction with any particular book in the series positively correlates to that book’s fairy tale structure as enumerated in Propp’s
system of 31 functions of a folktale’s dramatis personae.2 In other words, readers will report less aesthetic satisfaction the less concordance obtains between the tale’s actual morphology and Propp’s scheme; readers will report more aesthetic satisfaction the more concordance obtains between the tale’s actual morphology and Propp’s scheme.

Introduction

I. Propp and Potter

Vladimir Propp was one of the leading figures of the Russian formalist school of literary theory. His seminal work, the Morphology, was published in 1928, but not translated into English until 1958. By that time the winds of theory had shifted in other directions in both Russia and the West. Nevertheless, since the 1960s, this work inspired a number of studies in multiple disciplines among English-speaking scholars. Its applicability to the folktales of other cultures, to other kinds of folk narrative and performance, to non-
folklore literature and other cultural materials, and to the learning and transmission of fairy tale structure in children has been examined.3 One such study brought to my attention by a colleague examined the narrative schema in accounts of human evolution.4

The possibility of applying Propp’s schema to the Harry Potter series has been noted before. The first instance found in the literature is Joan Acocella’s review of the series up to the then-released Goblet of Fire.5 Acocello claims that Rowling was successful because of
her “utter traditionalism,” and Acocello proceeds to tick off a string of literary genres that many other readers have identified. She then connects this literary borrowing to Propp, whose schema of functions she deems a list of “just about every convention ever used in fairy tales.” She then lists six of the functions and fills them in with plausible story elements from Philosopher’s Stone. However, Acocello plays fast and loose with Propp’s work and especially the four morphological laws (see below), treating the functions precisely as conventions and without respect to the formal organization that is central to Propp’s scheme. A couple of sentences later Acocello equates Propp’s functions to archetypes, which is an unhelpful confusion of literary approaches.

In “Of Magicals and Muggles: Reversals and Revulsions at Hogwarts,”6 Jann Lacoss applies Propp’s schema to The Philosopher’s Stone and The Goblet of Fire. After a brief paragraph of introduction to Propp, Lacoss claims that “[t]he Harry Potter series seems to employ these same functions, although not always in the proper order (in the Harry Potter series, they are actually quite often in the same order as in magic tales).”7 She is quite correct here, as our own analyses confirmed (albeit with significant variation from Lacoss’ tables, which we will discuss below in “Evaluation”). She also correctly claims that “each book follows the sequence, and the overall plot of the series also appears to do so.” This is quite prescient given that Lacoss was working only with the series through Goblet of Fire. Furthermore, she speculates as we do that the language of tales “may be learned and
sublimated from childhood. Thus when the books were written, Rowling had an instinctive ‘road map,’ so to speak, for creating an engaging tale to which children (and adults) could easily relate.”8 Like Lacoss, we are not arguing that Rowling followed this structure intentionally. It is more likely that, like the Potter readers enthralled for reasons they know not, Rowling followed unconsciously the “cultural script” of folktales in writing the Hogwarts saga.

II. Morphology of the Folktale

Vladimir Propp finds that the magical folktales of his native Russia conform to a schema of thirty-one functions. He derived this schema from a systematic analysis done on a set of 100 stories in the collection of fairy tales compiled by Alexander Afanasyev. Propp produced symbolized representations for about 50 of those to demonstrate in abstract description the repetitive and uniform structure of these tales. These morphologies enable the folklorist to do comparative analysis within individual tales and among multiple tales.

Vladimir Propp finds that the magical folktales of his native Russia conform to a schema of thirty-one functions. He derived this schema from a systematic analysis done on a set of 100 stories in the collection of fairy tales compiled by Alexander Afanasyev. Propp produced symbolized representations for about 50 of those to demonstrate in abstract description the repetitive and uniform structure of these tales. These morphologies enable the folklorist to do comparative analysis within individual tales and among multiple tales. and their “amazing multiformity, picturesqueness, and color”12 (thanks to the settings, characters, objects, and other variables of infinite variety).

Propp argues that two further structural laws follow from his morphological study of tales: (3) that the sequence of functions is always identical and (4) that all fairy tales are of one type in regard to their structure.13 Since the sequential progression of functions is always the same, there develops a single narrative axis in all fairy tales. The position for a given function is always the same in every tale, though a particular function need not be present at all. These two morphological laws are central to our study and assessment of the Harry Potter series of books.

III. The Big Idea

[Commentary on (a) sequential structural analysis vs. Levi-Strauss’ paradigmatic structural analysis which organizes stories according to a matrix of paradigmatic thematic units, typically expressed in a set of oppositions; (b) Propp’s approach is isolated from the tale’s social and cultural contexts; (c) the approach is a way to begin to answer the question of HP’s popularity; (d) this approach is helpful because its methods are empirical and inductive, and the results here are reproducible by a similarly trained analyst; (e) archetypal analysis, Marxist criticism, feminist criticism, reader-response criticism, and other semantic-focused theories leap too quickly into genre analysis, authorial intent, and the social construction of the text without giving due consideration to the historical and sociological facts of the common cultural patterns that obtain in the narrative structure of folklore materials of all kinds in both Indo-European and non-Indo-European societies. Children all over the world hear many fairy tales; most like to hear them repeatedly. By the time they become readers the narrative sequence of familiar stories has been mapped onto their minds. Tales “go” a certain way. Specific actions should be present for it to “work” for the listener or reader. Irrespective of whether we determine the Harry Potter series of books as a fairy tale according to non-structuralist criteria and methods of analysis, we should not be surprised that they are structured like other popular stories in bestseller
fiction, comics, graphic novels, movies, and so on. If the Harry Potter series of books does not harmonize with Propp’s schema, then we would have to look for others reasons that readers are so easily and effectively drawn into the story (…)]

Methods

I. Aesthetic Satisfaction of Harry Potter Readers

An online survey was prepared and administered.14 We distributed the survey to students who had completed our Harry Potter course and to colleagues hosting Harry Potter-related academic and fan sites. Respondents first answered whether or not they had read all of the books in the series. Those responding “No” were discarded from the data set.
Next, respondents were asked the number of times they had read through the entire series. This indicates the probable familiarity the reader has with the story details of the series. These levels of familiarity are denoted as follows:

  • A Novice has read the series only once.
  • An Amateur has read the series more than once but fewer than five times.
  • An Aficionado has read the series more than five times but fewer than ten.
  • A Savant has read the series more than ten times.

Respondents then selected the rank order of the books in the series from least aesthetically satisfying (1) to most aesthetically satisfying (7).

II. Morphological Analysis of the Harry Potter Books and Series

According to Propp, “a tale may be termed any development proceeding from Villainy [A] or lack [a], through intermediary functions to marriage [W], or to other functions employed as a dénouement.”15 The analyst tabulates all of the functions in the tale and then summarizes the results in a symbolic string using Propp’s notation for the individual functions.

The 31 functions of the dramatis personae are organized in family units. Propp suggests his taxonomy of group-to-function can be likened to the biological relation of genus-to-species. Extending the biological metaphor, most functions have varieties which Propp denotes with numeric superscripts. The function with the largest number of varieties is Villainy [A] with 19. For the purposes of this study varieties of functions were not identified. The six genera of the functions are shown below with the corresponding species of functions that belong to each genus:

propp-schema

There is an Initial Situation [α] that is not counted as a function; it enumerates the family members or introduces the hero by name or status. The lower case ‘a’ in the Complication group denotes a “Lack,” which is an alternative form of Villainy [A] wherein a family member either lacks something or desires something important. The movement of the tale depends on the presence and type(s) of [A] or [a] function, but this pair is exclusive; a single narrative axis cannot have both an [A] and [a]. It is possible to have both an [A] and [a] if a tale has multiple moves. A move is created by a new Villainy (or Lack). These may be woven into the primary narrative axis either consecutively or concurrently.

Results

I. Aesthetic Satisfaction of Harry Potter Readers

Table 1 shows the response by mean score (on a scale of 1, representing least aesthetic satisfaction, to 7, representing greatest aesthetic satisfaction):

aesthetic-satisfaction-mean

Table 1: Summary of survey results showing the mean scores for aesthetic satisfaction

 

Table 2 shows the response by mode, which shows the score most frequently assigned by respondents to the given text. That is, a mode of ‘1’ indicates that ‘1’ was the most frequently assigned value out of all possible values ‘1’ through ‘7’.

aesthetic-satisfaction-mode

Table 2: Summary of survey results showing the mode for aesthetic satisfaction

The findings in Table 2 bear some further comment. We are interested in the extremities of the data. At the upper end of the range, the value indicating highest aesthetic satisfaction, both Prisoner of Azkaban and Deathly Hallows were chosen more frequently and consistently by readers as the most aesthetically satisfying tales in the series. At the low end of the range, the value indicating least aesthetic satisfaction, both Chamber of Secrets and Order of the Phoenix were chosen more frequently—though not consistently—as the least aesthetically satisfying tales in the series. For Novice readers, none of the books in the series achieved a mode of ‘1’; ‘2’ was the lowest mode and it was assigned to Chamber of Secrets. Aesthetic judgment hardens in the Amateur and Aficionado readers; two books indicate a mode of ‘1’ and two books indicate a mode of ‘7’. Two modes do not appear in their rankings (‘3’ and ‘4’ for the Amateur; ‘2’ and ‘4’ for the Aficionado). For the Savant, aesthetic judgment softens for Philosopher’s Stone and Chamber of Secrets. Only Order of the Phoenix retains a mode of ‘1’ for the Savant whilst Prisoner of Azkaban and Deathly Hallows retain their modes of ‘7’. Lastly, it should be pointed out that all readers consistently ranked Half- Blood Prince as ‘6’ more frequently than other scores.

II. Morphological Analysis of the Harry Potter Books and Series

We analyzed each book in the Harry Potter series to determine its concordance with Propp’s folktale structure. The narratives in each book were decomposed into their basic elements and these functions of the dramatis personae were identified and tabulated. After these tables were completed, the entire Harry Potter series was treated as one story to determine its folktale structure. All of the moves internal to the series were reduced so that the essential narrative axis could be determined and evaluated. The complete tabulated results of these analyses with narrative descriptions for each function are given in the Appendix. The symbolized representation of each book in the series and the series as a whole are given below. The basic components are arranged sequentially from left to right in their respective symbolized schemes. An asterisk denotes an out-of-sequence function or group of functions. The dénouement for all but Half-Blood Prince and Deathly Hallows ends with Harry’s return to 4 Privet Drive. We designate this with the italicized W.

1. Philosopher’s Stone (double-move tale)

philosophers-stone

The Reconnaissance–Complicity sequence in the upper move is Dumbledore as victim-hero and an independent axis is maintained throughout the tale. The Dumbledore axis terminates at a Solution [N]. The lower axis is Harry as seeker-hero. The original villainy done to Harry and his parents is out-of-sequence in this narrative and so is omitted in this scheme. Because the liquidation of this villainy is the primary movement driving not only Stone but the entire Harry Potter series, and because it is not resolved until Deathly Hallows, it will be the main villainy of the series of books taken as a single tale.

2. Chamber of Secrets (double-move tale)

chamber-of-secrets

The multiple Reconnaissance-Delivery moves include the ordinary type by the villain (top sequence) followed by two reverse types (Professor Binns is the middle sequence; Polyjuice Potion to get information from Draco is the bottom sequence).

3. Prisoner of Azkaban (no moves, but two heroes whose axes converge briefly)

prisoner-of-azkaban

The top narrative axis has Sirius as the victim- and seeker-hero. The bracketed Preparation and Complication groups for Sirius are disclosed near the end of Harry’s narrative. The bracketed Transference group < D E F G > denotes Sirius-Padfoot’s collusion with Crookshanks out of Harry’s sight. Harry’s Preparation group is demoted below the main axis because Sirius is the hero in Prisoner as borne out by the Q—W dénouement. The Victory [ I ] occurs in the Shrieking Shack after Sirius is vindicated and Harry shows mercy to Peter Pettigrew. The I—Rs sequence represents Sirius and Harry as temporary joint heroes. The Difficult Task [M] of rescuing both Sirius and Buckbeak (and as it turns out, Harry himself) is given to Harry and Hermione by Dumbledore. The Solution [N] includes an endless feedback loop to the earlier [Rs] because of the use of the time-turner.

4. Goblet of Fire (4-move tale)

goblet-of-fire

Goblet has the purest folktale structure. Its narrative scheme is the simplest representation of the seven books in the series. Harry is both the victim- and seeker-hero and he endures multiple villainies in four moves: three moves represent the three Tri- Wizard tasks and the final move is the direct conflict with Voldemort in Little Hangleton graveyard.

5. Order of the Phoenix (double-move tale with a spliced Transference sequence)

order-of-the-phoenix

The main axis concerns the villainy done to Harry and Sirius by Voldemort. The move below the main axis concerns the villainy of Umbridge and the Ministry. The out-of- order sequence in the main axis concerns the events from the founding of Dumbledore’s Army through Harry’s peering into Snape’s “worst memory” (Chs 16-28). The double-move is neither consecutive nor concurrent; such an incongruity in the linear sequence of the tale is a notable structural anomaly.

6. Half-Blood Prince (no moves, but two heroes whose axes converge)

half-blood-prince

The main axis is Dumbledore as the victim-hero; he was mortally wounded by the villainy of the Gaunt-Slytherin ring horcrux prior to the narrative’s beginning. Harry’s secondary hero status in this tale is indicated by his action halting at the second Difficult Task [M] where it joins with the Solution [N] now governed by Dumbledore’s actions. In the context of the entire Harry Potter series, this tale is further complicated by a third victim- and seeker-hero: Severus Snape. However, within the confines of Half-Blood Prince, his status as such is not recognized and is therefore omitted from this scheme. An interesting corollary study to the present one could examine Snape as the main hero in the Harry Potter series.

7. Deathly Hallows (7-move tale)

deathly-hallows

Although Deathly Hallows is arguably the most complicated tale in the series, none of its functions is out of sequence. Classifying its multiple moves is complicated by the carryover of the discovery of Voldemort’s horcrux creations and Dumbledore’s dispatch of the trio from Half-Blood Prince. Moves considered on the basis of Villainy [A] or Lack [a] internal to the Hallows narrative alone, there are four; considering the sequence of multiple Mediations [B] as moves, there are seven. Since these Mediation-moves lead to additional receipts of magical agents [DEF] for the purpose of completing the liquidation of all the preceding villainies, it is appropriate to consider these as moves in Deathly Hallows.

There are several notable sequences in Hallows. The first Complication includes the efforts to retrieve the real locket horcrux from Umbridge. Dumbledore’s bequest includes the donation of three magical agents to aid each member of the trio of heroes which are put to use at later points in the narrative. The next notable sequence is the combination of three interwoven sequences representing the efforts to acquire the means to destroy the retrieved locket horcrux, the sword of Gryffindor. The remaining horcrux-artifacts are retrieved and destroyed with less complication as exhibited by the consecutive single axes for each. The final sequence in the story, [D – W], occurs in the Forbidden Forest and the Battle of Hogwarts. The Epilogue ends with Albus Severus Potter absenting himself from home and the next chapter in the Hogwarts saga begins.

Scheme for the Series Treated as a Single Tale

series-as-a-whole

The Preparation group is the backstory developed and revealed in Prisoner of Azkaban, Half-Blood Prince, and Deathly Hallows. The Mediation [B], the misfortune made known to Harry, is gradually unfolded in the first six books of the series, and Dumbledore gradually dispatches Harry to defeat Voldemort. The Counteraction [C] occurs in Deathly Hallows: it is initiated at the beginning of the school year and it is confirmed conclusively by Harry at Dobby’s grave. Harry’s final test is in the Forbidden Forest with the Snitch, the magical agent donated to him by Dumbledore. Harry is revealed to have survived the killing curse in a reverse Exposure (Exrev). The return to the main axis is to complete the struggle with Voldemort and defeat him.

Evaluation

We now desire to assess the results of our analyses and come to some conclusions regarding our leading question.

Comparison to the Lacoss Schemes (2004)

First, we return to the tables for Philosopher’s Stone and Goblet of Fire prepared by Jann Lacoss.16 If we translate her table for Philosopher’s Stone into its symbolic notation, we get:
lacoss-philosophers-stoneA couple of critical notes are in order. First, Lacoss has the D and ↑ functions out of order in her table. Second, she identifies both a Villainy [A] and a Lack [a], but these are alternative types of one function and cannot appear in the same narrative axis. Third, an initial
Villainy [A] or Lack [a] is always liquidated [K], but Lacoss omits this essential function. Lastly, some of the assignations are dubious. For example, she has Branding [J] as “People notice [Harry’s] scar,” but the Branding function should be a consequence of the preceding combat between the hero and the villain [H]. Harry receives his scar in the prequel to the narrative and, at least in Stone, cannot be located at this point in the narrative. Lacoss’ complete list of descriptions is shown in an adjacent column to ours in the tabulated results in the Appendix. We believe the double-villainy directed at Harry within the narrative sequence of Stone is essential to describe correctly this story’s formal organization.

If we translate Lacoss’ table for Goblet of Fire into its symbolic notation, we get the following scheme:

lacoss-goblet-of-fire

Structurally, this is very similar to our reconstruction of this story (II.4). It is noteworthy that Lacoss’ moves occur on the Beginning Counteraction [C] move rather than on a Villainy [A] or a Lack [a]. However, if her assignment of the Villainy on the main axis were corrected (it should be an ‘A’ rather than a Lack [a]), picking up the axis at the struggle with Voldemort in the Little Hangleton graveyard completes that pairing at the Liquidation [K] function. The sequence (I K ↓) is misassigned to elements preceding the Struggle [H] in the graveyard. Lastly, the Unfounded Claims [L] function is misapplied; Lacoss simply names the false hero dramatis persona, but this does not fulfill the requirement for the narrative constant. Again, her complete list of descriptions is shown in an adjacent column to ours in the tabulated results in the Appendix. We conclude that Lacoss has correctly identified the general structure of Goblet, but has incorrectly analyzed some of the key sequences of functions in the primary axis of the story.

Assessment of Concordance Between Propp’s Scheme and the Harry Potter Series

We now turn to a comparison of our findings of the aesthetic satisfaction with the individual tales in the Harry Potter series and the results of our own complete narrative analyses. Propp’s approach is a data-driven investigation and description of folktale morphology; therefore, it is difficult to answer normative questions about whether a given story represents a “good” instance of the folktale structure or not. What are the criteria for making a meaningful judgment with the data a Proppian analysis yield? We can exclude some measures. It would not be meaningful to count the number of functions present in a Harry Potter story and use the full slate of 31 functions as a benchmark. For example, if Philosopher’s Stone (say) had 23 functions, it would not make any sense to infer that its concordance with folktale structure was therefore 75%. This is because tales often do not involve all of the functions (in fact almost none do), nor are they required to. Concordance with folktale structure does not depend on the maximal use of all the available functions in
Propp’s schema.

One possible basis for a measurement of concordance is out-of-sequence functions. Recall that Propp’s third and fourth morphological laws require that the sequence of functions is always identical and that all fairy tales are of one type in regard to their structure. A possible means of determining concordance with folktale structure, then, is to examine the number of out-of-sequence functions and the extent of their displacement from their “correct” position. We thus define a measure of incongruity by multiplying the two quantities. That is, Incongruity = No. of nonsequential functions × Displacement of nonsequential functions

If we tally the nonsequential functions and compute the Incongruity for each of the Harry Potter stories, we get the following results:

Summary-of-Discordance

Table 3: Summary of Incongruities in the Harry Potter Stories

Obviously, the value of Incongruity is relative; there is no scale or units of measure against which to interpret this quantity. So we must limit our interpretation to the most basic kinds of comparison. We see that Order of the Phoenix has the greatest Incongruity measured against the linear sequence of folktale structure. Philosopher’s Stone has the next greatest Incongruity. We recall that Order is also one of the Harry Potter stories that readers reported had the lowest aesthetic satisfaction, returning a mode of ‘1’ from all but Novice readers. We also recall that Stone was the second least aesthetically satisfying story according to its mean score. If we examine the other end of the spectrum, we see that Prisoner, Goblet, and Hallows had no Incongruity from folktale structure. We recall that Prisoner and Hallows both returned a mode of ‘7’ from all readers, and that they were the top two aesthetically satisfying books in the Harry Potter series according to their rank by mean score. We arrive at a general—and provisional—deduction: If a folktale has a high degree of Incongruity, the aesthetic satisfaction of the reader will be low. We cannot, however, deduce the contrary, for Chamber has a low Incongruity yet it is viewed as the least aesthetically satisfying story in the series. These are interesting results, but there are other measures for assessing the Harry Potter stories.

In the discussion which follows the presentation of the 31 functions of the dramatis personae in the Morphology,17 Propp makes several general “deductions” about the observed patterns which emerge from an examination of individual folktales “at close range.” He asks, “What does the given scheme represent in relation to the tales?” and answers thusly: “The scheme is a measuring unit for individual tales. Just as cloth can be measured with a yardstick to determine its length, tales may be measured by the scheme and thereby defined.”18 If Propp is correct, then we have confidence that in the current study we may therefore “define” the tales in the Harry Potter series and the series as a whole. We extract the following five propositions about folktale schemes that Propp identifies and discusses in the remainder of the Morphology. We will present them in the order they appear in his text.

Common Pair Arrangements

Proposition 1: “we observe that [the following functions] are arranged in pairs:”19

  • Prohibition—Violation [γ δ]
  • Reconnaissance—Delivery [ε ζ]
  • Struggle—Victory [H I]
  • Pursuit—Rescue [Pr Rs]

When we examine the schemes for the Harry Potter books, we find these notable results concerning their common pair arrangements:

  • Only Chamber has a Reconnaissance—Delivery [ε ζ] pair on its main axis that does not target Harry (it targets Ginny Weasley)
  • Only Goblet and Hallows have all of the pairs

There are notable results for other books in the series, especially Half-Blood Prince, but we will limit our assessment to the books identified earlier at the extremities of readers’ aesthetic satisfaction.

Proposition 2 concerns what tasks the hero is given or undertakes. Propp claims that “(…) it is always possible to be governed by the principle of defining a function according to its consequences. (…) all tasks giving rise to a search must be considered in terms of B; all tasks giving rise to the receipt of a magical agent are considered in terms of D. All other tasks are considered as M, with two varieties: tasks connected with match-making and marriage, and tasks not linked with matchmaking.”20 When we examine the schemes for the Harry Potter books, we find these notable results concerning their task differentiation:

  • Only Order has no M task whatsoever
  • The M task in Chamber is less urgent than the M tasks in the other books
  • Prisoner and Hallows have an M task that ends in a “match” or wedding
  • Only Prisoner has a B task with a victim- and seeker-hero other than Harry (Sirius)
  • Hallows has more than three D tasks (there are a whopping nine)

There are notable results for other books in the series, especially Stone, but we will limit our assessment to the books identified earlier at the extremities of readers’ aesthetic satisfaction.

Spheres of Action of the Dramatis Personae

Proposition 3: “(…) many functions join logically together into certain spheres. These spheres in toto correspond to their respective performers. They are spheres of action.”21 The following spheres of action are present in folktales:

  • Villain [A, H, Pr]
  • Donor [D, F]
  • Helper [G, K, Rs, N, T]
  • Princess (or sought-for person) [M, J, Ex, Q, U, W]
  • Dispatcher [B]
  • Hero [C, ↑, E, W]
  • False Hero [C, ↑, E, L]

When we examine the schemes for the Harry Potter books, we find these notable results concerning their spheres of action:

Only Prisoner has a villain other than Voldemort22

  • Hallows has a whopping seven Donors
  • In Hallows, the seeker-hero includes the whole trio and the victim-hero includes Snape
  • Snape is the false villain in Prisoner and Hallows
  • Draco is the false villain (by narrative misdirection) in Chamber
  • Only Order has a villain Donor (Umbridge)
  • Only Prisoner has unique Helpers in each Helper function (Fred & George Weasley, Dumbledore, Sirius, Harry, Hermione)
  • Order and Hallows have a clearly identified Sought-for person
  • Harry serves as a Dispatcher in Order and Hallows

Multiple Villainies, Interwoven and Sequential

Proposition 4: “A tale may be termed any development proceeding from villainy (A) or a Lack (a) through intermediary functions to marriage (W), or to other functions deployed as a denouement. Terminal functions are at times a reward (F), a gain or in general the liquidation of misfortune (K), and escape from pursuit (Rs), etc. (…) This type of development is termed by us a move. Each new act of villainy, each new lack, creates a new move. (…) One move may directly follow another; but they may also interweave (…).”23 We find these notable results concerning multiple villainies:

  • There are no multiple villainies in Prisoner or Hallows
  • The villainies in Chamber only indirectly affect Harry
  • Only Order has an interwoven villainy

Exclusive Pairs of Functions

Proposition 5: “(…) we observe that there are two such pairs of functions which are encountered within a single move so rarely that their exclusiveness may be considered regular, while their combination may be considered a violation of this rule (…). The two pairs are the Struggle with the villain and the Victory over him [H – I] and the Difficult Task and its Solution [M – N]. In 100 tales, the first pair is encountered 41 times, the second pair is encountered 33 times, and the two combined into one move three times [Some moves exist which develop without either of these pairs.].”24 We find these notable results concerning exclusive:

  • Only Order lacks an M—N pair
  • Chamber, Goblet, and Hallows break the exclusionary rule and have both pairs
  • Prisoner has a hybrid pair: the H-I pair involves Sirius on the primary narrative axis whilst the M-N pair involves Harry on the secondary axis

If we examine these results, we can deduce two outstanding patterns, one positive and one negative. First, we find several unusual features that become apparent when its structure and dramatis personae are related to the other books in the series and the series as a whole. We regard these unusual features as creative exploitations of the structural phenomena described by the Propositions.25 Second, we can identify clear violations of the Propositions. In the two books in the Harry Potter series that were identified as the least aesthetically satisfying, Chamber and Order, we find that there are fewer Creative Exploitations and more Violations than in the stories viewed as more aesthetically satisfying.

Chamber violates Propositions 1, 2, 3, and 5:

  • it has a Donor who is also a Villain
  • it has a Hero who is also a Dispatcher
  • it has an anomalous villainy move.

If we examine the two books in the Harry Potter series that were identified as the most aesthetically satisfying, Prisoner and Hallows, we find that there are fewer Violations and more Creative Exploitations than in the stories viewed as less aesthetically satisfying. Prisoner has no Violations of the Propositions. On the other hand, it has three Creative

Exploitations:

  • a B-task with a hero other than Harry
  • an M-task that results in a customary [W] function.26
  • a villain other than Voldemort.

Hallows has one Violation: it contains the exclusive pairs H-I and M-N. On the other hand, it has six Creative Exploitations:

  • it has all of the common pair arrangements (Goblet is the only other story in the series that has them all, too)
  • it has nine D-tasks, more than twice as many as any of the other stories (Stone is closest with four)
  • an M-task that results in a customary [W] function.
  • it has seven Donors (the tale with the second largest number of Donors is Prisoner with five)
  • Snape is revealed to be a false villain
  • Snape is revealed to be a victim-hero (and also a seeker-hero outside the explicit narrative of Hallows.

Conclusion

Let us summarize our assessment and conclude this study of the Harry Potter series. We recall that Propp regards the schema of 31 functions as a “measuring unit” for folktales. Our provisional conjecture was that one “measuring unit” for the discordance of any given tale to the schema is the quantity we have defined as Incongruity. We then arranged in five Propositions the “deductions” Propp derived from the formal organization of the schema. From our analysis and interpretation of these Propositions, we derived two additional qualitative measuring units: Violation and Creative Exploitation. We therefore have three independent measuring units with which to assess the concordance of the Harry Potter series of books to the fairy tale structure of the 31 functions of the dramatis personae: Incongruity, Violation, and Creative Exploitation. The properties of Incongruity and Violation are disconcordant; the property of Creative Exploitation is concordant. If we gather together our assessments of these properties determined in the previous section, and correlate them to the results we determined concerning the aesthetic satisfaction of readers, we get the following results:

concordance-and-discordance

Table 4: Measure of Concordances in the Harry Potter Stories

Recall that to interpret these findings correctly, the negative measures, Incongruity and Violation, correspond to relative disconcordance with the schema, whilst the positive measure, Creative Exploitation, corresponds to relative concordance with the schema. Therefore, Chamber and Order have relatively disconcordant fairy tale structures whilst Prisoner and Hallows have relatively concordant fairy tale structures. This correlates with the respective aesthetic satisfaction reported by readers. We therefore conclude that our hypothesis is confirmed: the aesthetic satisfaction with any particular book in the Harry Potter series positively correlates to that book’s fairy tale structure as enumerated in Propp’s system of 31 functions of a folktale’s dramatis personae.

The folktale structure is at least one of the keys to the success of the Harry Potter series. We believe that critics who dismiss the artistic value and meaning of the series by pointing out its alleged formulaic motifs, conventional mores and normative identities, arrive at this judgment erroneously and often with limited familiarity with the story. However, they are correct insofar as the Harry Potter series of books follows the schema of fairy tale as outlined by Vladimir Propp. This uniform and repetitive structure congruent to that found in the larger family of such tales accounts for its apparent formulaic character. Moreover, we find that readers’ aesthetic satisfaction with specific tales within the series correlates to those tales’ conformity to the fairy tale structure. If we want to answer the question “How did Harry Potter cast his spell over so many reader?” we need to account for the diversity of aesthetic responses to the individual tales in the series. The Harry Potter series of books cast its spell over readers by closely adhering to the formal organization of folktale structure. J. K. Rowling crafted an expansive fairy tale that has proven to be aesthetically satisfying to readers of all ages around the world. She successfully traced an ancient and enduring “cultural script” found in folk narratives we learn from childhood and articulated this structure with “amazing multiformity, picturesqueness, and color” in her settings, characters, magical objects, themes, and artistic virtues.

Acknowledgements

This investigation was undertaken with the invaluable assistance of my undergraduate research team at Barrett Honors College, Rita McGlynn, Tracie Smith, and Saswati Soumya, without whom this study would not have been possible. The author gratefully acknowledges their contributions to this study through their painstaking critical analysis of the Proppian functions in each book of the Harry Potter series and their review, corrections, and suggestions to this report.

Footnotes

    1. In this paper we will use the terms ‘folktale’ and ‘fairy tale’ interchangeably to describe the same type of literary unit, following the ambiguity inherent to the Russian skázka, the word used in Propp’s Morphology.
    2. Vladimir Propp, Morphology of the Folktale (1928), 2nd ed., trans. Laurence Scott, ed. Louis A. Wagner (Austin, TX: University of Texas Press, 1968).
    3. Alan Dundes, Introduction to the 2nd ed of the Morphology (1968), xiv-xv
    4. Pace John M. Lynch, the work is Narratives of Human Evolution by Misia Landau (New Haven: Yale University Press, 1991), x-xi; 3-16.
    5. Joan Acocella, “Under the Spell; Harry Potter explained,” The New Yorker (31 July 2000): 74-5. John Granger also noted Acocello’s article in How Harry Cast His Spell: The Meaning behind the Mania for J. K. Rowling’s Bestselling Books (Tyndale House, 2008), 21.
    6. In The Ivory Tower and Harry Potter: Perspectives on a Literary Phenomenon, Ed. Lana A. Whited (Univ of Missouri Press, 2004): 67-88.
    7. Ibid, 85.
    8. 85.
    9. Propp, 20.
    10.  He refers to four “theses” (pp 20-3) of which this is the first. It is clear within the pages of the Morphology that these “theses” function as invariant laws of structure. We therefore refer to them as Propp’s four morphological laws.
    11. Ibid, 21.
    12. 21.
    13. 22-3
    14. Appendix
    15. Propp, 92.
    16. Lacoss,86.
    17. Propp, 64ff.
    18. 65.
    19. 64-5.
    20. 67-8.
    21. 79-80.
    22. Although it should be noted that “Fear itself” is a reification of Voldemort’s persona.
    23. 92.
    24. 101-2
    25. As we noted above, Order has one such artistic exploitation, but three violations.
    26. Harry’s family is somewhat restored by his finding and uniting with his godfather Sirius. Only Hallows has the other customary [W] function in the series—Harry’s marriage to Ginny, and Hermione’s marriage to Ron.

The Relevance and Irrelevance of Heisenberg’s Uncertainty Principle for the Quantum Measurement Problem

This paper on the Uncertainty Principle has not been presented or published, but is drawn from my philosophical research into the Measurement Problem in quantum mechanics. A pdf version may be downloaded here, and my other papers on this subject are also available on my Academia.edu site.

The Relevance and Irrelevance of Heisenberg’s Uncertainty Principle for the Quantum Measurement Problem

Quantum mechanics is not something you would have guessed. The moment you juxtapose quantum mechanics and everyday experience, the mysteries of how the former relates to, much less explains, the latter seem to have no end. Scientists are predisposed to take the obviousness of the world for granted (rightfully so) while trying to explain and justify quantum mechanics. Many philosophers also take the obviousness of the world for granted (improperly so). But there are a few philosophers who have taken note that the very obviousness of the world is rather surprising. It’s surprising because that which is so obvious is at the same time so unobtrusive; it is so obvious it practically insists that we overlook it. Why does the world already make sense to us, at least in an unreflective way, the moment we turn our attention to it, before we’ve had a chance to formulate the first question about it? The child contends with and utilizes gravity long before its unceasing effects arouse curiosity. Upon a moment’s reflection, we can see that our first tentative intellectual steps toward understanding, like learning our first musical tune, are already upheld by a robust commitment to the consistency and congruity of sensuous experience. We enter the world with a basic commitment to the world, what Merleau-Ponty called “perceptual faith.”

Then why does quantum mechanics, the most empirically successful physical theory ever formulated, exhibit features that are inconsistent and incongruous with our understanding of the world, subverting our perceptual faith? Or do we exaggerate the strangeness of quantum mechanics? Is it in fact replete with everydayness? In my view, to “save the phenomena” of quantum mechanics from irrationalism, one must first recover the felt wonder of the world from its mundane unobtrusiveness. Perhaps by recovering the primordial sense of the world we would also find that quantum “mysteries” cohere with it. One of these quantum mysteries is the measurement problem. Before we turn to this problem specifically, we should briefly review the nature of measurement generally.

Measurement is a mathematical activity that constitutes the possibility of a thing not just being present in the here and now, but in a mode that is ever-present, identical for any subjective viewer. Thus, it is the primitive mathematical act. There are other such objectifying activities of a mathematical character (comparison, subordination, colligation), but measurement is that special activity whereby we are engaged concretely with a thing in the register of the sensible and we construe that thing in terms of a number. Measurement is of a higher order than the other basic mathematical activities of putting things side by side, or ordering them with respect to one another, or binding them together. These, too, involve us concretely with things, i.e., in the realm of the visible or the tangible. But measurement makes the number of a thing as definitive and provides the entry (some might say the escape) to a rarefied realm of intelligibility “beyond” the sensible.

The measurement problem in quantum mechanics

The broad foundational question about the connection between quantum theory and physical reality has an attenuated form in the so-called “measurement problem.” The problem is straightforward: when juxtaposing probabilistic expectations for experimental outcomes with real experimental experience, the world “shows a unique real datum: an actual fact.” One might be inclined to think that this is no more a mystery than how an actual number turns up when rolling a fair die. The difficulty is that the theory gives no account of how an actual unique datum comes to be realized at the end of an individual measurement (whereas ordinary classical mechanics tells how this works in the case of the die). Furthermore, when measurements are repeated under the same conditions, even repeated many times over, the disjunction between the theory and the actual outcome holds every time. “The status of actual facts in the theory remains nevertheless an open and troublesome question. Where does this uniqueness and even this existence come from? This is undoubtedly the main remaining problem of quantum mechanics
and the most formidable one.”1

Most scientists work under the presumption that doing more science will resolve this problem. But not all of them are willing to stride, like one of Arthur Koestler’s sleepwalkers, through such an admitted “fundamental obscurity.” John Bell is perhaps one of the more famous of these realists. He demanded that any interpretation of quantum mechanics meet the minimum condition of maintaining the Copernican perspective that displaced human beings from the center of the universe. Accordingly, he argued that concepts such as ‘observable’ and ‘measurement’ were “rather woolly,” and being anthropocentric, had no place in an authentic physical theory.

So, within the community of scientific practitioners we see fundamental disagreement over what the measurement problem means and what are the conditions for its solution and explanation. How shall we get our bearings? What is the convergent perceptual setting upon which theorists diverge conceptually? Specific examples are easy to find because the problem exists for any kind of quantum measurement, indeed all quantum measurements, whether the system is as simple as a single photon exciting an atom or as complex as the highly energetic experiments in giant accelerators searching for new particles. It seems that some measurements within this range of events should fail to have determinate outcomes. But this flies in the face of manifest perceptual experience. So how is the transition from indeterminate states to determinate states effected?

Quantum mechanics provides a set of causal principles which describe and predict the mechanics of a quantum system. The functional cornerstone of these principles is the unitary transformation postulate, which describes how one state at some initial time evolves into another state at some later time. The problem is that the foundational deterministic equation arising from this postulate (the Schrödinger equation) seems to exclude the possibility of a measurement ever occurring. So theorists add a separate principle of measurement, which requires a rupture to the smooth, linear evolution of the quantum system. This postulate requires the theorist to “project” what was a potentially determinate value onto an actually determinate value. It is this projection postulate that has sustained the most attention and criticism because it introduces physically incomprehensible notions like “collapse of the wave function” or “reduction of the state vector.” It is an admittedly irrational worm in an otherwise lovely apple.

What are we to make of a (purportedly) sensible thing that seems to have no definite place or position? If ‘to be’ means ‘to be there’, then how are we to understand the ‘there’ of a photon or an electron that is described by a wave that propagates everywhere? (Heidegger’s meditation on nearness and annihilation in the opening section of “The Thing” is appropriate here.) Furthermore, how are we to understand an object that is not indifferent to acts of observation or measurement? How must we transform our classical view of measurement, that a pre-formed reality is open to human observation while yet remaining uninfluenced by actual measurements, in the light of quantum mechanics where measurement is an intrinsically invasive procedure?

A phenomenological analysis of the measurement problem would involve at least five questions:

  1. Why has measurement become a “problem” in quantum mechanics? What is the true source of the trouble?
  2. How do the entities investigated “phenomenalize?” How do they emerge into the register of the sensible, the visible?
  3. How does measurement of quantum entities and processes relate to phenomenalization?
  4. What is the role, if any, of human involvement, e.g., perception, in measurement?
  5. What is the role, if any, of human involvement in the thing measured? What does the measurement problem teach us about the (non)sensible thing?

In this paper, I shall limit our considerations to the first question. Let us begin by setting to the side any predetermination of the “reality” or “existence” of the entities in question. We need to minimize the influence of our natural predispositions to talk about and think about atoms, electrons and photons as if they were ordinary things like tables and chairs, an equivalence which is manifestly not the case. My phenomenological feint is not proposed in order to answer the same questions taken up elsewhere, only now from a “phenomenological” perspective, whatever that might mean to the hearer. What I do hope to elucidate is the nature of the watershed in physics between realists and anti-realists, its genealogy, and other possibilities that might be envisioned.

In order to do justice to the task of concrete research, I begin from an atypical beginning than most philosophical research on the subject of quantum mechanics. I want to gain some understanding of how the measurement problem ever became a problem at all. This is not meant in the sense of a question of empirical history. Rather, we will need to undertake some conceptual archeology. The measurement problem, characterized as an interaction between an observer and something observed, suffers from the obscurity created by the entrenched conceptual doublets of modernist metaphysics (nature-man, mind-body, self-other, subject­object, constituting agent-constituted thing, etc.). The first task then, is to clarify the interaction

at the root of the measurement problem on a basis that is not so conceptually hamstrung. It is true that there are many claims in the scientific and philosophical literature that the measurement problem has been solved (or that it is merely a pseudo-problem), but the proposed “solutions” entail other nonrealistic consequences (e.g., nonlocality); and, while these insights are philosophically suggestive, so far, solutions to the measurement problem have merely transposed the original problem into a different register with the same metaphysical precommitments.

Heisenberg’s uncertainty principle and the Pythagorean root of quantum mechanics

Why has measurement become a problem in quantum mechanics? Because it, more than the other presumed problems of quantum physics, is the problem of foundations. It is a philosophical problem posed by physics. Other features of quantum mechanics which have incited much philosophical reflection are not our real concern, even though they sometimes are falsely associated with the measurement problem. Chief of these is Heisenberg’s famous “Uncertainty Principle.” This tenet of quantum mechanics, which to many is so closely associated with the inherent “mystery” of quantum theory, is not, as it turns out, relevant to quantum mechanics per se. It is quite simply not a discovery or determination unique to quantum mechanics. It tells us little to nothing about the concrete aspects of microscopic phenomena and our involvement with them (despite breathless claims to the contrary in some popularizations of quantum mechanics, beginning with Heisenberg himself). The “Uncertainty” (better: “Indeterminacy”) Principle is a mathematical artifact created by a precommitment to economical priorities in the interest of simplifying calculation or computability, not from measurement disturbances. The Heisenberg indeterminacy relation takes two forms:

Δp Δq > h/4π (1)
ΔE Δt > h/4π (2)

They express the variance of two canonically conjugate2 quantities: momentum and position in the first case and energy and time in the second case (Heisenberg’s original derivation published in 1927 describes an electron moving in empty space). The right side of the inequalities is a constant, with Planck’s constant, h, in the numerator (6.626 x 10-34 J • s). This indeterminacy principle is as ubiquitous as potsherds in the mathematical sciences. Let us examine the ways in which it appears in different guises in the theoretical and applied sciences and attempt to trace its genealogy.

In my undergraduate days in electrical engineering, I toiled long hours on signal analysis. Be it an osprey call or an FM radio transmission, all signals have two elementary features, irreducible (though transformable) to one another: time and frequency. In the case of the bird song, you can listen as the signal varies in intensity through time. You can also hear, at any given moment, the pitch or pitches of the signal, its frequency component. You cannot hear all of the frequencies simultaneously, just like you cannot see all of the colors in ordinary light; you need a tool to break up the complex signal into its components. For light, we use a prism; for signals (more precisely: for the functional representation of a signal), we use the Fourier transform. When you transform a signal from the time domain into the frequency domain you transform a signal into a spectrum with harmonics at different frequencies and different magnitudes (amplitudes). Now, a signal generated “naturally,” or “in the wild,” is sloppy; the tones aren’t pure or perfect, they’re “noisy.” The clicks, chirps, chattering, or other interruptions to the subject of the signal (speaking musically) are not the features that we want to stand out; quite the contrary, we want to filter them out so that the subject stands out more clearly.

In the representation of the signal, there is always some spread or variance from where the frequency is centered, the value around which it is concentrated (in statistics, this is the expectation value). This is where the indeterminacy relation enters: there is always a minimum degree of divergence between the two spreads, between the time variance and the frequency variance, and that divergence is expressed as an inequality:

s x S ≥ 1/16π2, (3)

where s is the time signal variance and S is the Fourier transform or frequency signal variance.

What does this particular mathematical expression mean? The purer (or clearer or more defined) the time signal, the fuzzier is the frequency signal. And vice versa: the clearer the spectrum, the more indistinct the time signal. Note well: the indeterminacy (or “fuzziness”) is
not an aspect of the actual phenomenon as it is experientially manifest (e.g., the osprey call that I hear); it is a result of the abstract analysis we have applied to the signal, which is a functional representation of the phenomenon (in the case of the bird song, a representation of something audible). In other words, the mathematized expression of the bird song re-presents an irrevocable distortion of the original phenomenon, the song as it is sung or heard. But how did we generate this mathematical artifact? Hidden within the function we applied to the signal to determine the variances s and S is a simplification: it is linear. But the original signal to which we applied the function is nonlinear—there is harmonic distortion, frequency compression, clipping—and vastly more complex than we would prefer or manage for calculative purposes. So, for economic reasons, we make a simplification, we make the math more convenient. Note well: other interests shape and guide us, practical interests, according to which we discard features or elements of the phenomenological totality for the sake of aesthetic, pragmatic or other considerations.

Now, it is no accident that the Heisenberg indeterminacy equations (1) and (2) have more than a family resemblance to the signal variance inequality (3). Structurally, they express the same relation: the product of two spreads or variances on the left side which is greater than or equal to some constant value. And, just as was the case with (3), we must keep in mind that (1) and (2) are also functional representations of abstract concepts; i.e., the indeterminacy of ‘position’ or ‘momentum’ spread expressed by the equations is not a feature of a concrete phenomenon. Furthermore, these are also ideal operations: the resolution of one variable can be varied infinitely with corresponding deterioration or improvement in the quantitative determinacy of the companion variable without any implication that some real sound in the world approximated by one of the variations is itself sensuously indeterminate. Where theorists too often go astray is in the common assumption (since Galileo) that mathematical phenomena transparently and unproblematically map onto or correspond with the phenomena encountered and engaged in experiential manifestness, “in the wild,” if you will; that our neat, cultivated idealities must have some positive ontological status, either in themselves or as the only “true” representation of some concrete phenomenon. Obviously, this selective perspective or eidos of bird songs, electromagnetic waves and electrons means that the way in which we are going to contrive these as objects and signify the world itself as object is as rigorously representable by linear means.

Let me give another example to reinforce my earlier claim about oft-overlooked simplifications of linearity. One of the most popular mathematical expressions formulated in the twentieth century is E=mc2. This is Einstein’s famous mass-energy equivalence formulation, a follow-up to his original Special Relativity theory of 1917. The relation between energy E and mass m is modified by a constant of proportionality, c2. But, there are an infinite number of nonlinear terms on the right side of the equation that are not shown that make a more precise determination of the desired variable (either E or m) far less manageable. Exactitude is sacrificed for elegance. It is no wonder that the trade is sought given the high value placed on an objective sense of balance (viz., laws of conservation) and completeness and totality of representation. Thus, the determination of a number by measurement does not entail that precision or exactitude of quantity is the desired aim.

Modern natural science finds the pragmatic principle of “for all practical purposes” indispensable. Analyses are condensed or abridged without noticeably relinquishing control of prediction, planning or common standards of measurement. Some conscientious theorists are uneasy with this pragmatic incursion into quantificational matters because they can find no rational basis for calling a halt to what they already know is rational, viz., the mathematical rigor of the formulation and the certitude of the calculative operations. Can we know in advance the value at which we’ve reached the threshold of mathematical “control”? Why or why not? If so, can we state or specify this a priori as clearly and distinctly as the mathematical certitude derived from it? If not, is there anything from “nature” other than experimental repetition or a posteriori empirical operations that we can point to as a basis for our decision to interrupt the infinite iterations that unfold before us? These are the questions that need to be asked and that constitute the real philosophical import of the Heisenberg indeterminacy principle.3 But, historically speaking, we could have asked these questions before quantum mechanics was formulated.

Heisenberg indeterminacy relations, both quantum and classical, arise because the world is just too complex, or, speaking mathematically, “nonlinear,” for our practical purposes. In the process of idealization from nonlinear to linear, and abstraction from phenomena “in the wild” to their more docile, cultivated mathematical representations, we simplify the representations of concrete phenomena so that we can perform linear math on them. We find ourselves in a forest out of a Brothers Grimm tale and in order to make sense of it, we raze, prune, trim, flatten, and straighten all the wildness out of it until we have a tame, formal, English garden. The “higher order” terms are ignored as Rococo excesses of nature. This sweeping approximation requires
the insertion of an estimated value into what is manifest (the “knowns”), a straightening of crooked curves and wiggles, a smoothing of rough terrain, i.e., an idealization. No matter how disheveled the crown of a tree, one can always determine smoothness by arbitrarily narrowing
the focus to a smaller region. What must always be borne in mind is that the “global” view of the tree manifestly differs from the linear, smooth, local view. I am not inferring that we cannot thereby mathematicize the global phenomenon; I merely wish to point out that we ought to avoid recklessly transferring the “good fit” of a linear formulation from a local level to a nonlinear holistic level.

We need to trace the ancestry of indeterminacy relations still further, for we have not yet reached their origin. Indeterminacy relations are found throughout the mathematical sciences, both classical and quantum. The quantum indeterminacy inequalities (1) and (2), and my chosen example of a classical indeterminacy inequality (3), are both representations of abstract objects: variance in ‘position’, ‘energy’, ‘frequency’, etc. Once objectified, these conceptual abstractions can be thought of in some (abstract) space. How are they related to one another in this abstract space? The fundamental mathematical activities (e.g., comparison, subordination, colligation, measurement) are not available to us in a non-sensible register, so we require a higher order analysis. First, we represent magnitudes by fixing arbitrary points A and B in an abstract space. Arbitrary vectors, A and B, can then be drawn with lines from a common origin to the two points. To complete their relation, construct an orthogonal (perpendicular) projection of one line to be superimposed on the other line. This projection creates a right-triangle relationship and leads to the “normal” equations of least-squares curve fitting. This projected right triangle contains a Heisenberg indeterminacy relation, the Cauchy-Schwarz inequality, which relates the lengths of the two vectors (the product of their norms) to the absolute value of the inner (or dot) product (also called the ‘correlation’) between them:

lA x lB ≥ |A ● B|, (4)or‹x, y›2 ≤ ‹x, x› ● ‹y, y› (5)

in generalized bra-ket notation. The Heisenberg indeterminacy principle just is the quantum mechanical expression of the Cauchy-Schwarz inequality. But recall how we generated the Cauchy-Schwarz inequality: two abstract straight lines and the formation of a right triangle. This procedure allows us to “normalize” unfixed vectors and simplify the “fitness” of an unknown quantity given a minimum number of known quantities. The paradigmatic example of determining an unknown value in light of two known values is the solution of the length of the side of a right triangle or the deflection of one of its unknown angles. All of the mathematical sciences, including both quantum and classical physics, insofar as they utilize or impose the constraint of linearity, contain an indeterminacy relation whose common ancestry can be traced to the Pythagorean theorem.

So, the mathematical formalism of quantum mechanics—its abstract “objects” (operators) and “space” (Hilbert space)—finds its roots not only in the algebraization of geometry begun by Descartes (quantum mechanics makes extensive use of the linear algebra generalized from analytic geometry) but also in the humble beginnings of Pythagorean and Euclidean geometry. Indeterminacy relations ultimately rest on the ubiquitous Pythagorean theorem. Underlying the modern use of the Pythagorean theorem is a notion of problem-solving and optimality whereby an unknown path is inferred from known components. The theorem depends on orthogonality conditions whereby two abstract objects intersect as if they were the legs of a right triangle. The orthogonality conditions permit the easiest way to find a trend in a scattering of data points and filter some of the noise from your car radio. The application of the Pythagorean theorem outside the realm of pure geometry, the finding of an optimum direction or value, the simplest interpolation, the easiest or least calculation, all indicate the supremacy of a principle of economy. But that is most certainly not a Pythagorean or Platonic principle. A philosophical reorientation was required to make it possible to have an interest in simplifying a problem for
calculative purposes.4

Linearization is achieved by application of the Pythagorean theorem and it enables us to focus our efforts on the elements of a system that matter for calculative control, on the determination of manageable parts. This is precisely the approach taken in quantum mechanics. The initial appearance of subatomic entities and electromagnetic radiation as classically wavy phenomena allowed theorists to study them using well-understood and relatively simple concepts of linear wave mathematics: reflection, diffraction, interference, intensity, frequency, periodicity, superimposition. Formally, quantum mechanics is not about “things in the world” but about swarms of “linear operators” in a cosmos of matter waves. But how are we to understand the necessary interface between these classically derived concepts, these abstract objects, this
abstract space, and the perceived world of lived experience where these formulations are confirmed, the empirical manifold, the world of manifest perceptual experience?

Linear mathematics spawns indeterminacy relations. So, our philosophical interest is spurred not by indeterminacy relations per se, but by their origin in linearity assumptions. If a system is nonlinear, the parts of the linearized subsystem do not add up to the whole. The behavior of groups cannot be sufficiently understood as the accumulation of their components’ behaviors. Even if the mathematical artifacts of linearity assumptions, the indeterminacy relations, are somehow transferable or superimposable on phenomena, then it is possible to ask: how can it be that a measuring instrument (or a measurer, for that matter), which is a big, complex chunk of material, is a reliable guide for studying the finest divisions of matter? This question remains highly controversial and the analysis of indeterminacy relations can carry us no further. We must seek out the question where it is questionable, not in the register of the intelligible entities of mathematical operations, but in the register of the sensible. This is the central problematic for further phenomenological research on the measurement problem.

  1. The Interpretation of Quantum Mechanics. Roland Omnès. (Princeton, NJ: Princeton University Press, 1994), 60­
    61, 350.
  2. ‘Canonically conjugate’ variables are “quantities that are not independent of each other,” i.e., they have some relation such that one is irreducible to the other.
  3. It is on these foundational questions that you find commendable philosophical sensitivity on the part of physics theorists in the scientific literature.
  4. An excellent review of this tectonic shift is found in David Lachterman’s The Ethics of Geometry: A Genealogy of
    Modernity. See also Marc Richir’s excellent review of this book.

Greetings!

This is the academic and professional site of Dr. Joel Hunter. This is the home of his academic and scholarly work while at Barrett Honors College at ASU, the University of Kentucky, and Georgetown College, as well as his professional work as an environmental and electrical consulting engineer. My current project on this website is a multi-part tutorial on how to succeed in an Honors Seminar.