Evaluating al-Ash‘ari’s Change of View on Religious Epistemology

As I said before, I am going to try to publish some of the papers I wrote in the past on this blog so I could (1) have a more complete record of my thoughts in one place, and (2) better organize my thoughts, iA. This was a paper I wrote back in May, 2012 for RELI 3559, a course at U.Va. on Islamic theology taught by R.B. Siebeking. Unfortunately, I have never thought about this topic again as deeply as I had at the time, but my sentiments remain the same: We need to bring back more rationality in Islam. P.S. Eid Mubarak! 🙂


May 7, 2012

Evaluating al-Ash‘ari’s Change of View on Religious Epistemology

The decrease of emphasis on the use of rationality in engagement with the scripture and the traditions, the replacement of theology with jurisprudence as the dominant focus among the scholars, and the transition from ijtihad to taqlid in the Islamic intellectual tradition in the medieval period all seemed to have been voluntary moves to block intellectual progress and seemed to have thereby contributed indirectly to the development of the Salafist movement that so shaped 20th century Islam. As part of my effort to understand this paradigm shift in religious epistemology, I would like to examine a key figure from the earliest period of this transition—Abu al-Hasan al-Ash‘ari, theologian and eponym of the Ash‘arite school of theology. Although little is known about al-Ash‘ari’s life, his “conversion” from Mu‘tazilism to the more conservative traditionalism (ahl al-hadith) is widely known and constituted a significant episode in the history of the Islamic intellectual tradition. What was the nature of his conversion? Why did he change his view regarding the use of reason? Unfortunately, historical sources do not provide clear answers. It is not even possible to compare and contrast al-Ash‘ari’s views pre- and post-conversion, as barely any of his pre-conversion works have been preserved. In fact, out of the supposed some hundred works of al-Ash‘ari, only around six have survived. In this paper, I will thus content myself with evaluating al-Ash‘ari’s post-conversion view on reason in considering these three questions: 1) Did his change of view make sense in the historical context? Did the historical context provide him with any motives for converting? 2) Is there any intellectual continuity between a general Mu‘tazilite view on reason and his later view? Is there rational consistency in his later view? 3) What impact might his views have had on the scholars of later generations and on the broader Islamic intellectual tradition, with reference to the question of epistemology? The bulk of this paper will be devoted to the evaluation of al-Ash‘ari’s view in consideration of the second question.

The Historical Context

During the third/ninth century, following Caliph al-Ma’mun’s campaign of religious persecution known as the mihnah, “a strong reaction set in against the ‘rationalist’ kalam” of the Mu‘tazilite court theologians. Van Ess suggests that this “ordeal” (meaning of mihnah) probably aroused in the general public a certain kind of negative collective memory about the Kharijites, who had permanently linked any practice of takfir with exclusivist extremism[1]. In addition to this violent act, which considerably discredited their image, the Mu‘tazilites were also holding a position that was becoming increasingly irrelevant to the contemporary situation. Watt postulates that “Mu‘tazilism was essentially an attempt to work out a compromise that would in part overcome the cleavage between Sunnites and Shi‘ites.” However, by the middle of the ninth century, this effort had evidently been abandoned by the government in favor of a pro-Sunnite regime, as both Sunnism and Shi‘ism irreversibly solidified. Meanwhile, people had become more disillusioned with the government and the theology it promoted. Eventually, the Mu‘tazilites became no more than “a group of academic theologians who had retired to an ivory tower remote from the pressures and tensions of the times,”[2] and indeed far away from the hearts of the common people.

Replacing the rationalist trend was the rise of the “Hadith folk”[3], who were the strict followers of the Hadith and the jurists (notably, Imam Ahmad ibn Hanbal) who resisted the use of reason and accepted the scripture and the traditions literally, “without asking how” (bila kayfa)[4]. Such complete subordination of reason was diametrically opposed to the Mu‘tazilite view, which affirmed the primacy of reason in the attainment of religious knowledge. It was in this climate of contradiction, in 874, that al-Ash‘ari was born. Despite belonging to an orthodox family, al-Ash‘ari studied under and became one of the best disciples of the Mu‘tazilite master of Basra, al-Jubba‘i (d. 915). However, at around age 40, he publicly renounced Mu‘tazilism during a Friday prayer and thereafter devoted his life to defending his new orthodox position which in many ways represents a middle path between the two extremes.

Did his conversion make sense given the historical context? Although the reversal of his opinion could appear sudden and arbitrary to the observer, when considered in the historical context as described above, it would not seem so surprising anymore. Extreme positions like Mu‘tazilism and Hanbalism simply could not be held for an extended period of time without incubating, in the concerned thinker, some desire for reconciliation. In fact, al-Ash‘ari was neither the only nor the first person to pursue “rationalist orthodoxy”. Before him, ash-Shafi‘i already “held that there should be a certain number of men trained thus to defend and purify the faith, but that it would be a great evil if their arguments should become known to the mass of the people”[5], and two other contemporary theologians, at-Tahawi (d. 331) and al-Maturidi (d. 333), were also starting to work in the same line of thought. MacDonald speculates hence that the inclination to reconcile reason[6] and orthodoxy, albeit being subconscious, was already there in the historical current; people just didn’t recognize its existence.

This “subconscious” theory could even account for the change within al-Ash‘ari himself. A famous report of how al-Ash‘ari converted is that he dreamed of the Prophet three times (some psychologists say that dreams contain representations of the dreamer’s subconscious thoughts): “Twice the prophet said, ‘Help the traditional beliefs.’ After the second command he gave up theology and spent his time on the Koran and tradition. In the third vision Muhammad asked what he had done and then said, ‘I did not tell you to drop theology but to defend the traditional beliefs for they are the truth.’”[7] And that’s when al-Ash‘ari finally realized the new position that he needed to defend.

Regardless of what actually happened, what we can be sure of through an understanding of this historical context is that al-Ash‘ari’s conversion was most likely not a result of a sudden decision but a gradual transformation; a transformation that was caused in part by a general political dissatisfaction with the Mu‘tazilite position and the influence of the Hadith folk and the popular thoughts. However, perhaps the most important reason for his conversion was his intellectual and theological dissatisfaction with the Mu‘tazilite doctrine. This claim can be supported by the comprehensiveness, complexity, and acuity of the theological position found in al-Ash‘ari’s works. Someone who just decides to convert based on sentimental waves simply does not construct such sophisticated and rationally consistent theories. In the next section, I will examine one of these theories—his theory about the place of speculative reasoning in theology—and try to determine why he might have been motivated to formulate this new view on reason.

A Rational Evaluation of al-Ash‘ari’s ‘ilm usul ad-din

Although Al-Ash‘ari’s ideas might very well have been developed consciously or subconsciously under the influence of the historical current, as suggested in the last section, the most explicit reason for his conversion is generally recognized to be his intellectual dissatisfaction with the Mu‘tazilite position. For, an account even more famous than the dream narration is the “three brothers” story, in which al-Ash‘ari raised an objection disproving the Mu‘tazilite doctrine of “the best” (al-aslah)[8] that silenced his teacher al-Jubba‘i:

The story is that he asked his master: “What do you say of a believer, an unbeliever, and a child?” Al-Jubba‘i replied: “The believer is in heaven, the unbeliever in hell, and the child in a place of safety.” Al-Ash‘ari asked again: “But should the child ask God why he did not let him grow up that he might earn a bigger reward?” Al-Jubba‘i: “God would say that He knew that he would be a sinner if he grew up.” Al-Ash‘ari retorted: “The unbeliever would ask why God did not kill him that he might not sin.” Al-Jubba‘i had no answer.[9]

The story has an ironically similar quality to that of the Mu‘tazilite founding story involving Wasil ibn ‘Ata withdrawing from the circle of al-Hasan al-Basri. The difference is that in the former, the intellectual dissatisfaction was with the use of intellect itself. It is a protest on al-Ash‘ari’s part, “a recoil from the impossible task of raising a system of purely rationalistic theology”[10] to reliance upon what is physical and unquestionable: the scripture, the traditions, and the practice inherited from the early Muslims.

So what did al-Ash‘ari now think should be the role of reason? Did he retain the use of reason in theology or did he abandon it completely like the Hanbalites, whose position he professed to hold? If he did retain reason, how big a role did he think reason should play? Although there has been much debate on this topic since the time of al-Ash‘ari himself[11], the rough general consensus is that he did use reason, but only insofar as it restricted itself to the scripture and the traditions. But this position seems to be problematic in the same way that saying “It never makes sense to say it never makes sense” is problematic; namely, that reason, in this case, is referring to itself in a way that prohibits or brings trouble unto itself. Does al-Ash‘ari’s theory manage to avoid this trouble? And if it does, are there other problems with his theory?

To find the answers to these questions, I will examine al-Ash‘ari’s view on the role of speculative reasoning in theology as presented in the prefatory section of his Risalah ila ahl al-thaghr bi-Bab al-Abwab (Epis­tle to the People of the Frontier at Bab al-Abwab) with the help of Richard Frank’s analysis of the piece. This risalah is a compendium of the usul ad-din on al-Ash‘ari’s view, composed shortly after his conversion, and in its preface, al-Ash‘ari discusses his “‘ilm” usul ad-din (the science of the fundamentals of religion) —essentially, his methodology for obtaining theological knowledge. It is important to note here the fact that he even elaborates on a methodology at all is something which is anathema to the Hanbalites, who only discuss the usul ad-din and do not go beyond them to formulate a method of knowing the usul ad-din. Basically, al-Ash‘ari’s methodology consists in following through a “rational order of the progression of faith” to obtain faith in the authority of the scripture and the traditions and thereupon obeying all the commands in the scripture and in the traditions without questioning. Here, it is obvious that following through the “rational order of the progression of faith” is the extra step of “kalam” that that the Hanbalite position lacks and disapproves of[12]. What is this rational progression of faith?

Essentially, al-Ash‘ari divides the teaching of the Prophet into four major topics and ordered them in a logical way so that the resulting sequence explains how one obtains faith in the authority of the Prophet. According to this conception, “the Prophet called his auditors to the faith by leading them through progressive stages of understanding until they came to recognize the origin and nature, and so the authority of his teaching.”[13] These four stages are as follows:

1) To acknowledge the contingency of the world and that of one’s own being

2) To acknowledge that the world and each individual are subject absolutely to the will of an omnipotent and provident God

3) To recognize that Muhammad is a true emissary of God

4) In consequence of this, to believe all that the Prophet says is to be believed and to do all that he has commanded to do[14]

The first three stages could be reached by applying reason to the arguments provided by the Prophet, and the fourth stage—acceptance of the authority of the Prophet sc. of the scripture and the traditions—is just a necessary moral consequence of having been through the first three. Basically, reason is a tool that we use to realize that we should do all that the Prophet has commanded us to do (without using reason). Another way to think about it is that the three stages are “evidence” for the validity of the fourth stage—the “command”; and you only need to use reason for understanding the evidence, not for following the command. Before analyzing the issues with this theory, I will elaborate on al-Asha’ri’s explication of these four stages.

What does the “evidence” look like? Basically, for the first two stages, the evidence are arguments (hujaj and adilla) used by the Prophet in the scripture and the traditions. The evidence for the third stage is the miracle of the Prophet, i.e. the Qur’an. Here are some of the arguments al-Ash‘ari says the Prophet used for the first two stages. For stage one, the acknowledgment of the contingency of oneself and of the world, al-Ash‘ari indicates that the Prophet pointed out to the people of the world the contingency of their being “through the alternation of form and disposition that occur in their persons and through the differentiation of languages”[15]. For the second stage, the acknowledgement of one’s subjection to God and God’s attributes, al-Ash‘ari says that the Prophet made known to the people of the world “the way to recognise their maker through the evidence found in themselves and elsewhere that requires His existence and demonstrated his will and His Providence.”[16] Although these particular arguments seem a little vague, al-Ash‘ari asserts that all the arguments presented in the canonical sources are rational, conclusive and better than any arguments the philosophers can come up with (for example, the arguments for contingency based on the idea of  “accidents”). And therefore reason itself, for its own perfection, should demand “its self-restriction to these arguments and conclusions.”[17] He also bolsters his claim by the historical sociological evidence that the first few generations of Muslims never asked for any more evidence than these.

As for stage three, the recognition that Muhammad is a true emissary of God, al-Ash‘ari claims that the veracity of the Prophet is established by signs and miracles. Muhammad had a special sign just as Jesus and Moses each had their signs (the miracles they performed), and that special sign is the Qur’an. However, in order to recognize that it’s a miracle, one needs to have been through the first two stages. One could thus call it a “conditional” miracle. Recognition of this miracle should then further confirm the validity of the first two stages, prove the veracity of the Prophet (the third stage), and also lead, necessarily, to the fourth stage: belief in and action upon all of the Prophet’s commands. While the connections between the first three stages are supposedly logical, the connection between the third and the fourth, here, seems to be one of moral necessity.

Now, here are three possible objections to this theory relating to the “self-referential trouble” that was alluded to earlier. While al-Ash‘ari might have answers to some of them, I think he could not provide an answer to at least the last one.

First objection: Granting that the rational order of the progression of faith is actually rational and that the evidence are actually conclusive[18], in order to obtain the evidence and the arguments for achieving the first three stages, one still needs to consult Islamic sources like the scripture and the traditions. Why does al-Ash‘ari seem to insist then that the authority of the scripture and the traditions is not the point of absolute beginning?

What al-Ash‘ari would say: It’s true that there can be no awareness of the Prophet’s claims and precise teachings apart from the transmitted sources, but these teachings in the beginning do not have “authority”; they are merely presented as evidence so that one would accept later the authority of all the claims and teachings as a whole. In other words, some statements from the transmitted sources are arguments and thus evidence for the first three stages, and some are just commands that one has to follow after one has achieved the fourth stage. One has to be careful in distinguishing between the two kinds. The authority of the scripture and the traditions, therefore, is not the absolute beginning. The absolute beginning is the arguments and the evidence contained within the scripture and the traditions.

Second objection: But how exactly does one distinguish between the evidence statements and the command statements? Although the Prophet might actually have presented to his companions the statements in the “rational” order that is suggested here, the fact is that after the first generation of Muslims, all we have are the texts—the scripture and traditions—without the “presenter,” who should have been the Prophet. Today, the “evidence” statements and the “command” statements are all jumbled up together in the texts. How are we supposed to distinguish the two kinds? How are we supposed to become convinced by the “evidence” statements in the first place when they are obscured by the other supposedly unreasonable “command” statements?

What al-Ash‘ari would (probably) say: Practically speaking, the problem of determining the criteria for distinguishing between the two kinds of statements simply does not arise. As one reads through the texts, one’s reasoning faculty would become so content with the evidence statements as a result of their perfect coherence with “the nature of things as given to human understanding in the experience of being”[19] (innate metaphysical knowledge), that we would eventually accept the entire religion, along with the command statements. In other words, one learns “the way to use the reports of the Prophet as a demonstration”[20] for the truth of all the claims of the Prophet. The knowledge of the difference between the two kinds therefore is simply a kind of a priori knowledge.

Third objection: So, the idea is that, even though there are still questions that remain from the command statements, the metaphysical and physical picture that has been spelled out by the evidence statements (automatically picked out due to their coherence with our reason) is already so complete and so attractive, that the person who scrutinizes these texts should be willing to and in fact will necessarily accept the whole package, including the points that are unclear. However, it seems that according to this theory of the rational order of the progression of faith, recognizing the contingency of created beings, God’s attributes, and the Prophet’s authority are the only things whose truths should be determined by the evidence statements (or perhaps that the evidence statements in the texts can only affirm the truths of these things), and everything else should be accepted as is, bila kayf. One cannot help but wonder, though: why should we only apply reason on these particular categories? Why can’t we simply apply reason on all the statements in the texts, and leave only the very ambiguous ones without asking how? What is really preventing us from using reason all the way through in interpreting the texts if the texts are in fact the truth?

I am not certain that al-Ash‘ari has an answer to this objection. However, the later Ash‘arite theologian Fakhr al-Din al-Razi did build on al-Ash‘ari’s theory and devised a way to deal with this problem by dividing the verses of the Qur’an into three types: those that have an apparent sense that can be confirmed by rational indicators, those that do not have an apparent sense that can be confirmed by rational indicators, but can be confirmed through other indicators, and those which are ambiguous both in meaning and indicators. “The sound exegete, according to al-Razi, will know how to discover the truth concerning the first two types of verses, and will know to entrust the meaning of the third type to God.”[21] Note, however, that al-Razi considers here only the Qur’anic verses and not the hadith literature, which probably contains too many contradictions for any appropriate application of reason. This observation is enlightening when considered in terms of al-Ash‘ari’s motive for conversion. It is possible that the crux of the whole matter was simply an unease al-Ash‘ari felt with the tension between rationality and certain contradictory commands contained within the hadith literature.

What has been presented so far in this section is a “rational” examination of al-Ash‘ari’s position on the role of reason in theology; that is, an assessment using reason, as a Mu‘tazilite might have done. The result of this assessment suggests that al-Ash‘ari did have some coherent reason for switching his view, namely that the arguments in the texts for the authority of the texts are conclusive and intellectually satisfying, and therefore all that is left for us to do is to accept the texts bila kayfa, as the Hanbalites preach. We can conclude hence that there is at least some intellectual continuity between the Mu‘tazilite view and al-Ash‘ari’s view on reason in that al-Ash‘ari did not simply abandon the use of reason but sought to subordinate it to the extent that it made room for certain irrational commands.

However, how could he justify his ‘ilm usul ad-din i.e. the use of kalam to the Hanbalites? We do not have the space to go deep into this issue here, but in his Al-hathth ala al-bahth (The Exhortation to Investigation), which is often published under the title Risala (fi) istihsan al-khawd fi ‘ilm al-kalam (A Vindication of the Science of Kalam), al-Ash‘ari defends the use of kalam by arguing that the Hanbalite argument for “kalam is haram” fails because, for one thing, the Prophet was not ignorant of the use of kalam. It only appears so because the language of kalam did not develop until later. In fact, all the arguments in the canonical sources are completely rational and can be readily translated into the language of kalam if need be. He gives many examples of what he believes to be these kinds of arguments and states:

All the verses which we have mentioned, as well as many which we have not mentioned, are a basis and argument for us in our kalam on what we mention in detail. It is true that no question was particularized in the Book and the Sunna. But that was because the particularization of questions involving rational principles did not take place in the days of the Prophet. However, (he and) the Companions did engage in kalam of the sort we have mentioned.”[22]

Although the Hanbalites would probably reject this argument outright because the argument itself is apparently the result of speculative reasoning, since we’re assessing the rational consistency of al-Ash‘ari’s position, the Hanbalite’s irrational rejection should not be a concern here. In any case, al-Ash‘ari did not seem to identify with Hanbalism at the end, even if he did in the beginning. And, as history has shown, al-Ash‘ari’s theology has proved to be a completely new school of thought which carved out a new path between Mu‘tazilism and Hanbalism and which became the mainstream orthodoxy.

To summarize, the Mu‘tazilites thought they could apply reason on everything, while the Hanbalites thought they should just accept everything without using reason. Al-Ash‘ari, out of his dissatisfaction with the inability of reason to deal with certain issues, put forward a new position advocating that one should use reason in order to accept the authority of the Prophet, but should not question any commands after this point. Although it was a good effort to bring together the two extremes, overall, I find that al-Ash‘ari’s withdrew from the use of reason too fast, too much. The theology that he constructed is still closer to the Hanbalite position and retains much of its intellectual inflexibility.

Impact on the Islamic Intellectual Tradition

In order to assess the impact of al-Ash‘ari’s religious epistemology on the broader epistemological practice of the Islamic intellectual tradition, it is necessary to understand the different types of religious epistemology. The typology I will present here is from Norman Calder’s essay “The Limits of Islamic Orthodoxy”. Calder suggests that “all possible forms of religious belief can be caught under the following five headings: scripture, community, gnosis, reason, charisma.”[23] “Scripture” is just the idea that God reveals himself through a set of written texts. “Community” means that God’s revelation to man is primarily through the community, “that one particular community has been chosen by God and within that community correct belief will be articulated and preserved, because that community is guarded by God and preserved from error.”[24] “Gnosis” denotes the process of obtaining divine knowledge through mystical experience, and is usually associated with the Sufis. What is meant by “reason” is straightforward; it is usually associated with two groups: the Mu‘tazilites and the philosophers. “Charisma”, on the other hand, is exclusively associated with Shi‘ism (except in certain Sufi circles), as Shi‘ites believe that God has appointed one person with special charisma to preserve His message.

Calder’s own analysis of Sunni orthodoxy is that Sunnis lie somewhere between scripture and community (or that they encompass both categories) while being closer to community. The scriptural texts, according to him, are “the residue of salvation history”[25] that constitutes authority in the religion. Then, after the period of salvation history ended, the Muslim community derives over the course of the rest of the history an orthodox theology and a system of law that are explained and justified by reference to the authoritative scriptural texts “through engagement in the exegetical process”[26]. And it is because this exegetical process is based on an intellectual tradition grounded in mutual reference and recognition within a scholarly community across time, that the Sunni Orthodoxy is judged to have a mainly community-based epistemology. McAuliffe, in her essay “The tasks and traditions of interpretation”, depicts how much Muslims scholars paid respect to past scholars even if what they really wanted to do was to make their own commentaries on the scriptural texts[27]. In fact, if a Muslim scholar made any commentaries based on his own reasoning or imagination, he would be marked as heretical. In his conclusion, Calder judges Sunni Islam to be “primarily a religion of community, scripture and gnosis, marginally of reason, and hardly at all of charisma.”[28]

Although this uneven distribution leaves something to be desired (e.g. more rationality), one remarks that there is certainly not a lack of diversity of epistemological categories defining the intellectual tradition of Sunni orthodoxy, and I believe that al-Ash‘ari did much to contribute to this diversity. During the time of al-Ash‘ari, Mu‘tazilism represented the epistemological category of “reason”, and Hanbalism represented (roughly) “scripture”. What al-Ash‘ari brought forth, as he created a theology from existing elements and thoughts—a theology that used rational arguments but was at the same time orthodox and acceptable to the community at large—was essentially the tradition of “community” to tie together both “reason” and “scripture”. As we have seen, al-Ash‘ari’s theology was not a perfect synthesis, but it did open the door to a more eclectic intellectual tradition, which was later epitomized by the celebrated Ash‘arite theologian al-Ghazali (d. 1111).[29]

In conclusion, al-Ash‘ari’s change of view on reason has had a relatively positive effect on the Islamic intellectual tradition. How much, then, has al-Ash‘ari’s theology contributed to the Salafi literalist movement that is characteristic of the Islam we know today? I would say: not much. The scholars that contributed the most to this movement were Ibn Taymiyya and fundamentalists in the modern period like ibn Abdul Wahhab. While al-Ash‘ari expanded the epistemological categories defining orthodox Islam, these literalists ignored both community and reason and indeed gnosis, reducing Islam to a much smaller tradition based only on the Qur’an and the sunnah. At first glance, this might not necessarily be a bad thing and could even have the effect of purifying the religion. However, when fourteen centuries of intellectual tradition had abruptly been cut out, the scriptural texts now existed only in a vacuum—their associated context being centuries removed from, and no longer relevant to the current context. In my personal opinion, this was an unfortunate turn. Perhaps what we need today is another al-Ash‘ari-like figure to reform and broaden our scope of religious epistemology.


Bibliography

Calder, Norman. “The Limits of Islamic Orthodoxy.” In Intellectual Traditions of Islam, edited by Farhad Dftary, 66-86. New York, London: I.B. Tauris & Co. Ltd in association with Ismaili Studies, 2000.

Frank. M. Richard. “Al-Ash`ari’s conception of the nature and role of speculative reasoning in theology.” In Early Islamic Theology: The Mu`tazilites and al-Ash`ari, edited by Dimitri Gutas, 136-154. Ashgate, 2007.

Gardet Louis and Anawati M.-M. Introduction à la théologie musumane : essai de théologie comparé. Paris : Librarie Philosophique J. Vrain, 1970.

Hodgson, G.S. Marshall. The Venture of Islam: conscience and history in a world civilization. Chicago: University of Chicago, 1974.

McAuliffe, Jane Dammen.“The tasks and traditions of interpretation.” In The Cambridge Companion to the Qur’an, edited by Jane Dammen McAuliffe. Cambridge University Press, 2006.

MacDonald, Duncan B. MacDonald, Development of Muslim theology, jurisprudence, and constitutional theory. Charles Scribner’s Sons, 1903.

McCarthy, Richard J., S.J., trans., “Abu al-Hasan al-Ash‘ari’s Risala (fi) istihsan al-khawd fi ‘ilm al-kalam.” In The Theology of Al-Ash‘ari, 119-134. Beirut: Imprimerie Catholique, 1953.

Nasr, Seyyed Hoseein. Islamic Philosophy from its Origin to the Present: philosophy in the land of prophecy. Albany: State University of New York Press, 2006.

Sands, Zahra Kristin. Sufi Commentaries on the Qur’an in Classical Islam. New York: Routledge, 2006).

Tritton, A.S. Muslim Theology. London: Published for the Royal Asiatic Society by Luzac, 1947.

van Ess, Josef. The Flowering of Muslim Theology, trans. Jane Marie Todd. Cambridge, London: Harvard University Press, 2006.

Watt, Montgomery W. Islamic Philosophy and Theology. Edinburgh: University Press, 1962.


Footnotes:

[1] Josef van Ess, The Flowering of Muslim Theology, translated by Jane Marie Todd (Cambridge, London: Harvard University Press, 2006), 31.

[2] Montgomery W. Watt, Islamic Philosophy and Theology (Edinburgh: University Press, 1962) 83

[3] Marshall G.S. Hodgson, The Venture of Islam: conscience and history in a world civilization (Chicago: University of Chicago, 1974), 438.

[4] Seyyed Hoseein Nasr, Islamic Philosophy from its Origin to the Present: philosophy in the land of prophecy (Albany: State University of New York Press, 2006), 124.

[5] Duncan B. MacDonald, Development of Muslim theology, jurisprudence, and constitutional theory (Charles Scribner’s Sons, 1903), 187.

[6] “reason” here means the use of intellect independent of scripture not grounded in taqlid, not “scriptural” reasoning.

[7] A.S. Tritton, Muslim Theology (London: Published for the Royal Asiatic Society by Luzac, 1947) 166.

[8] Namely, that God is constrained to do that which may be best and happiest for His creatures.

[9] Tritton, Muslim Theology, 166 (with modification).

[10] MacDonald, Development, 190.

[11] Richard M. Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning in theology,” in Early Islamic Theology: The Mu`tazilites and al-Ash`ari, ed. Dimitri Gutas (Ashgate, 2007), 137.

[12] We will see at the end of this section how al-Ash‘ari defends the use of kalam

[13] Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning in theology,” 138.

[14] Ibid.

[15] Abu al-Hasan al-Ash‘ari, Risalah ila ahl al-thaghr bi-Bab al-Abwab, p.82,2, quoted in Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning  in theology”, 138

[16] Abu al-Hasan al-Ash‘ari, Risalah ila ahl al-thaghr bi-Bab al-Abwab, p.82,2, quoted in Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning  in theology”, 138

[17] Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning in theology,” 144.

[18] …which could be problems themselves, but I will not raise the issue in this paper.

[19] Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning in theology,” 139.

[20] Abu al-Hasan al-Ash‘ari, Risalah ila ahl al-thaghr bi-Bab al-Abwab, p.90,7, quoted in Frank, “Al-Ash`ari’s conception of the nature and role of speculative reasoning  in theology”, 140

[21] Zahra Kristin Sands. Sufi Commentaries on the Qur’an in Classical Islam (New York: Routledge, 2006), 60.

[22] Richard J. McCarthy S.J., trans., “Abu al-Hasan al-Ash‘ari’sRisala (fi) istihsan al-khawd fi ‘ilm al-kalam,” in The Theology of Al-Ash‘ari (Beirut: Imprimerie Catholique, 1953), p.129,20.

[23] Norman Calder, “The Limits of Islamic Orthodoxy,” in Intellectual Traditions of Islam, ed. Farhad Dftary (New York, London: I.B. Tauris & Co. Ltd in association with Ismaili Studies, 2000), 71.

[24] Ibid.

[25] “salvation history” is just a history that defines a religion.

[26] Calder, “The Limits of Islamic Orthodoxy,” 77.

[27] Jane Dammen McAuliffe, “The tasks and traditions of interpretation,” in The Cambridge Companion to the Qur’an, ed. Jane Dammen McAuliffe. (Cambridge University Press, 2006).

[28] Ibid., 83.

[29] Louis Gardet and M.-M. Anawati, Introduction à la théologie musumane : essai de théologie comparé (Paris : Librarie Philosophique J. Vrain, 1970), 67.

Advertisements
Posted in Islam | 3 Comments

The term “revert” has no basis in Islam

This post is basically an expansion of a comment I made here as a response to Sheikh Abu Yusuf Riyadh ul Haq’s lecture “Commonly Misunderstood Verses of the Quran” back in 2011. Since the link on Halaltube is now defunct, here is a new link on YouTube. Apparently, it is an excerpt from Lesson 183 of his lecture series “al Tajrid al Sarih” (The Abridged Saheeh al Bukhari). Unfortunately, I cannot find the original lecture online.

The thesis of this post is that people are not born into Islam and the term “revert” has no basis in Islam. Among English-speaking Muslims, it is common to refer to someone who converted to Islam as a “revert”. The rationale behind the usage of this term is based on the hadith: “Every child is born upon fitrah.” Most Muslims interpret this hadith to mean that every child is born a Muslim and therefore believe that when a person becomes a Muslim, he/she is “reverting” to Islam. As the Wikipedia page on fitrah states: “… some Muslims prefer to refer to those who embrace Islam as reverts rather than converts, as it is believed they are returning to a perceived pure state.” Even the Oxford Dictionary defines “revert” as “A person who has converted to the Islamic faith.”

However, as Sheikh Abu Yusuf Riyadh ul Haq very clearly explains in his lecture:

Commonly, this hadith is used to prove that all people are born in the state of Islam, as Muslims, and therefore all children remain Muslims… That’s incorrect. This hadith doesn’t mean that every child is born a Muslim… The correct understanding of this hadith is that every child is born upon nature. And the natural state of every human being is one of clarity, purity; of being in a pristine form; of being unpolluted, unadulterated by external factors. And that natural state is conducive to a person embracing Islam. That natural state is one in which a person is inclined to the beauty and the truth of Islam, but not exactly a Muslim. As that child grows up, the external factors influence that child’s nature. If the external influence is corrosive and corruptive, then that purity would be affected, and the person would be disinclined to Islam… and thus they will become followers of other faiths or of no faith.

As emphasized here, “fitrah” means “the pristine, natural state of a human being that is conducive to belief in Islam”; it does not mean “Islam”. Therefore, people are not “born” Muslims. The “Islam” section on the Wikipedia page on “religious conversion” states: “Islam teaches that everyone is Muslim at birth”. This is simply not true. People are not born into Islam; they are born into a natural state of being that is one of purity and one that is conducive to belief in God.

“To revert” means to return to a previous state, condition, or practice, and Muslims generally understand it to mean “reverting to Islam”, based on the hadith under question. However, since fitrah does not mean Islam, we can hardly call someone who decides to be a Muslim later in life a “revert”. Perhaps one could argue that those people had “reverted” to that natural state of being before accepting Islam. But is that really true? Even if it is true, it is much more straightforward to call them simply “Muslims”, without any qualifiers. Note that all of the first generation of Muslims had converted to Islam from other faiths (or no faith at all). Did they call themselves “reverts”? No. They called themselves Muslims, plain and simple.

In my humble opinion, it is also best to try to avoid using the term “convert”. Technically, you’d have to have a religion before “converting” to a different religion. So, if you did not have a religion before becoming a Muslim, then you are technically not a “convert”. Even if you did convert, what is important is not the fact that you converted to Islam but the fact that you are now a Muslim.

If you do want to emphasize the fact that you were not raised in a Muslim family and came to accept Islam through other means, then IMHO it is better to say “I became a Muslim / accepted Islam X years ago / in year XXXX”.

Tl;dr: The usage of the term “revert” is based on the hadith “Every child is born upon fitrah”, commonly interpreted to mean every child is born a Muslim. However, since “fitrah”≠Islam, people are NOT born into Islam, and people who have converted to Islam should not be called “reverts” but simply “Muslims”.

Posted in Islam | Tagged , , | 5 Comments

A Mathematician’s Lament

I have recently read a little book: A Mathematician’s Lament by Paul Lockhart. Originally a 25-page essay, it was expanded into a book by popular demand and published in 2009. The author is a K-12 mathematics teacher who has a PhD in mathematics and used to teach at Brown University and U.C. Santa Cruz.

Mathematics as an Art

The central thesis of the book is that mathematics is an art, and should be taught like one. Mathematicians, like artists, are makers of patterns, except that they all follow the unifying aesthetic principle of “simple is beautiful“. Since the simplest possible things do not exist in the physical reality but in our imagination, and things that exist in our imagination are called ideas, it could be said that mathematicians make patterns with ideas, as the mathematician G. H. Hardy said in this quote:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. —G. H. Hardy

Just like art, there is no ulterior practical purpose to math. It is simply something that is fun to do, an activity that is done for the inherent pleasure derived from performing the activity. Mathematicians start by making up imaginary beings like triangles, then they ask questions about their intrinsic properties and behaviors. The art of mathematics lies in crafting satisfying and beautiful explanations (i.e. proofs) in response to these questions.

The major difference between ordinary art and mathematics, I feel, is that in ordinary art, you can control every aspect of your creation, whereas in mathematics, once an entity is born out of your imagination, what it then does is beyond your control; that would need to be discovered. And it is, I believe, through making such discoveries that mathematicians get to have a glimpse of divine beauty, which beyond any kind of beauty that human beings alone could create. This possibility to access divine beauty makes mathematics a higher form of art than any other.

To put it another way, although human beings are able to create imaginary beings in our mind, the patterns and structures in which our imaginary beings are embedded already exist out there in some “mathematical reality”. In this way, imagination is actually our personalized wormhole, our dokodemo door, to that fantastical world. Please note that this is my opinion, not the author’s. As expressed elsewhere on this blog, I am both a mathematical Platonist and a theist.

By the way, one can also look at this through a systems lens. I am currently reading Thinking in Systems by Donella Meadows, who defines a system as consisting of elements, interconnections, and a function or a purpose. If mathematics is a system, then its elements (numbers, spaces, etc.) would be those things that could be made up by the human imagination. But the interconnections among these elements (theorems) exist in the mathematical reality and would need to be discovered. As for the function/purpose of the system, well, that may be beyond our understanding at the moment!

The Horror of Math Education in the U.S.

Back to the book. Lockhart’s lament is about the status of math education in the United States. In the American classroom, the entire process of doing mathematics (wherein lies the “art”) is reduced to memorizing facts, following procedures, and mindless manipulating symbols. As Lockhart repeatedly writes: “The question has been asked and answered at the same time.” There is nothing left for the students to do.

Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity—to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. —p.29 A Mathematician’s Lament

By removing real mathematics from the math curriculum and presenting a hollow shell of the real thing, many potentially gifted mathematicians end up hating the subject. Lockhart goes so far as to say “Better to not have math classes at all than to do what is currently being done.” This reminds me of the principle: “First, do no harm.”

I love this quote (p.33, my bold):

I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers—the kind of thing a real mathematical education might provide.

I chuckled when I read that because I used to be that member of the society. As a typical victim of American math education, I had such a bad impression of the subject that I did not take a single course in mathematics when I went to college. Why would I? It was only until a few years after I graduated, when the bad memories had sufficiently faded and my interest in the subject had increased again thanks to my passion in logic and philosophy, that I had gathered enough courage and motivation to go back to college to study mathematics. Thankfully, I had wonderful professors the second time around (especially Professors Elwood Parker and Rudy Gordh, both now retired). They taught much much better than my high school teachers ever did. I am not blaming my high school math teachers. They were wonderful people. But unfortunately, they had to follow a particular curriculum that had little to do with real mathematics.

It is interesting to note that Lockhart included a whole chapter on high school geometry. To be honest, I liked that class a lot, especially the proofs part (despite the rigidity of their format). Math had never been my strong subject in school, but that year I was actually picked by my teacher to enter into a geometry math competition. As the author says on p.81: “It’s hard to completely ruin something so beautiful; even this faint shadow of mathematics can still be engaging and satisfying.” Unfortunately, my interest in math subsided again after Precalculus and AP Calculus. I would not see math again until seven years later.

Math is Useless and That’s OK!

I also love the author’s emphasis on the inherent uselessness of mathematics. Sure, mathematics has lots of practical applications, but these are just a trivial by-product of mathematics, and not what mathematics is about.

Algebra is not about daily life, it’s about numbers and symmetry—and this is a valid pursuit in and of itself. —p.38

Isn’t it wonderful to see that written out?

Now hold on a minute, Paul. Are you telling me that mathematics is nothing more than an exercise in mental masturbation? Making up imaginary patterns and structures for the hell of it and then investigating them and trying to devise pretty explanations for their behavior all for the sake of some sort of rarefied intellectual aesthetic?

Yep. That’s what I ‘m saying. In particular, pure mathematics (by which I mean the fine art of mathematical proof) has absolutely no practical or economic value whatsoever. 🙂

—p.120 (my bold and my smiley)

The Ideal Math Teacher

So what does Lockhart think should be done instead? How should mathematics be taught? Since I have been job-hunting for some time now (unsuccessfully, I might add… if anyone knows of an accounting position in NC or MO, please contact me on LinkedIn) and read more job postings than anything else, I thought it would be interesting to write a job description of the K-12 Math Teacher according to Lockhart’s ideals (I made up the numbers for years of experience and salary, lol).


K-12 Math Teacher – Entry Level

American public schools are looking for talented and passionate teachers to conduct daily lessons in Math.

Responsibilities (c.f. p.43):

  • Choose engaging and natural problems suitable to the students’ tastes, personalities, and levels of experience
  • Give students time to make discoveries and formulate conjectures
  • Help students to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism.
  • Be flexible and open to sudden changes in direction to which their curiosity may lead.
  • Have an honest intellectual relationship with students and with mathematics.

Qualifications:

  • Open and honest
  • Have the ability to share excitement
  • Have a love of learning
  • Have enough of a feeling for mathematics to be able to talk about it in your own voice, in a natural and spontaneous way

Experience:

  • Proofs (done by self): 1+ year
  • Field experience in Mathematical Reality: 1+ year

Job Type: full-time

Salary: $35,000 – $40,000 per year


Lockhart believes that “There should be no standards, and no curriculum. Just individuals doing what they think best for their students” (p.82). In other words, teachers should have the freedom to do what they think is best. Generally, professors are granted that freedom, and I think that is why mathematics at the university level is so much more fun.

By the way, here is the proper way to teach techniques (p.42):

Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.

This would seem counter-intuitive to many people, since most of us are inflicted with the Do Something Syndrome. But it is important to do nothing sometimes. Teaching is not transmitting information; it is giving students just enough guidance so they can learn by themselves. Here, I am reminded of the importance of white space in Chinese landscape paintings. If the entire paper is covered by a mountain, you wouldn’t be able to appreciate the majesty and beauty of the mountain. In the same way, if a question is asked and answered at the same time, students would not be able to appreciate a mathematical theorem for what it  is. Struggling in ignorance provides the white space that is necessary to appreciate the mountain that is mathematics.

After all, isn’t the universe born out nothingness? The void is the beginning of many things. OK, someone needs to stop me from philosophizing.

My Question for the Author: Role of Definitions?

Overall, I mostly agree with the author. However, he has a viewpoint that I am still not sure I understand, and that is “…you don’t start with definitions, you start with problems” (p.79). Here is his explanation (p.79-80, my bold):

In an effort to create an illusion of clarity before embarking on the typical cascade of propositions and theorems, a set of definitions is provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem generated. To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.

The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.

“Starting with problems” is probably how most professional mathematicians work, and seems to be the natural way for mathematics to develop. However, as a student I find definitions to be really important, and I rely a lot on them. Most of my professors seemed to think so too, and made us memorize definitions verbatim. Almost every math textbook I had in college follow this format:

  • Definitions
  • Theorem
  • Proof
  • [Repeat]

So even though I understand Lockhart’s argument conceptually, I cannot say that I truly understand the practical implication of his suggestion. Maybe he only means to attack definitions that are not necessary (as in the following quote found on p.58)? If so, then I would understand.

No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They are equal, for crying out loud. They are the exact same numbers, and have the exact same properties. Who uses such words outside of fourth grade?

That last sentence made me lol.

Or perhaps he is only complaining about the “timing” for introducing definitions? Maybe he is just saying that definitions should not be introduced at the very beginning, but only after students have struggled with problems themselves and have an initial understanding of the problem’s structure (much like how techniques should be introduced to students)?

Ending Remarks

Finally, I just want to say that I love all the little proofs in the book. They are a perfect demonstration of the exquisite elegance and the miraculous nature of mathematics.

I agree with Keith Devlin in his forward to the book (p.11-12):

In my view, this book … should be obligatory reading for anyone going into mathematics education, for every parent of a school-aged child, and for any school or government official with responsibilities toward mathematics teaching.

I don’t belong to any of these groups of people, but I still enjoyed the book. So I think if you are reading this, you will too!

Posted in Mathematics, Non-fiction, Philosophy | Tagged , , | 6 Comments

Analogy, perspectives, understanding, and imagination

This post is an attempt to summarize and expand on what I have already written on analogy and perspective by clarifying the relationship between the following concepts: analogy, perspectives, understanding, and imagination.

What is an analogy

Making an analogy involves two steps:

  1. Abstracting a general pattern from one or several instances, and
  2. Applying that pattern to a new instance.
making an analogy.jpg

Source of the original graph: https://www.youtube.com/watch?v=NVE8CaKsPPo

Step one (abstracting a pattern) by itself is essentially inductive reasoning.

Step two (applying the pattern) is looking at something from a particular perspective.

(I wonder if we can say that step two by itself is essentially deductive reasoning.)

The relationship between analogy and understanding

Let’s call the new instance we want to understand the “target”. The more you use analogy and apply different patterns to the target, i.e. the more you look at the target from different perspectives, the deeper your understanding of the target (I mentioned this idea here when discussing how seeing more instances of a concept perfects the concept).

The importance of having multiple perspectives

I already wrote about the importance of having a perspective. To begin to see/understand something, first you need to have at least one perspective (what I called a central belief). After you have a good view of the target from that perspective, you can then proceed to view it from a different perspective. But to see anything at all, first you must stand still.

Once you have a central belief, however, it is important to look at your target from different perspectives in order to gain a complete understanding of the target. Sometimes, it is impossible to have a direct understanding of something because we are not physically equipped for direct understanding (e.g. the parable of the blind men and an elephant and “seeing’ a tesseract, as discussed toward the end of this post). In that case, looking at the target from as many perspectives as possible is the best we can do.

Recap: analogy is the core of cognition

It is obvious now why analogy is the core of cognition. Analogy is the process of abstracting patterns and applying patterns. Applying patterns to something is the same as looking at something from a particular perspective. And the more perspectives you take to look at something, the deeper your understanding of that thing. Cognition is the process to obtain understanding. Hence, analogy is the core of cognition.

The role of imagination / Using analogies in mathematics

What role does imagination play in all this? In order to “look at” a target from different perspectives, we have to use our mind’s eye, i.e. we have to use our imagination.

So, the ability to understand is essentially the ability to to change perspectives, and such ability requires the faculty of imagination, as elaborated in this Ted Talk:

I am reminded of how the mathematician David Hilbert, upon hearing that one of his students had dropped out to study poetry, was quoted saying “Good, he did not have enough imagination to become a mathematician.”

It is interesting to note that both mathematicians and poets use imagination extensively, but their targets of understanding are different. Mathematicians use their imagination in abstracting and applying patterns in the most general sense possible for its own sake, while poets do the same thing (with their extensive use of metaphors) but only to the extent to capture and understand the human condition.

Here is another video on using analogies in math (I love how he explained the Euler’s identity this way!):

The link between imagination and understanding/knowledge also reminds of me of the famous Einstein quote: “Imagination is more important than knowledge. For knowledge is limited, whereas imagination encircles the world.”

I suppose this is so because imagination is part of the means to obtain knowledge. To use an analogy (ha): imagination is one machine in the cognition factory, and knowledge is what is produced by the factory (i.e. products). The number of products is necessarily limited by the amount of material (experience and data points) available to the factory, but the machine itself has unlimited potential to produce.

Patterns = Mental models (in the sense used by Charlie Munger)

I would just like to point out that what I mean by patterns here is the same as the concept of mental models as described by Charlie Munger. This is more of a note to self, since this is a concept that is very familiar to me. I have been a longtime subscriber of Farnam Street and I am currently reading 好好学习 by 成甲, who is also an advocate of the mental models theory.

Summary

  • making an analogy = abstracting patterns + applying patterns
  • applying patterns to something = looking at something from a particular perspective
  • look at something from different perspectives = understanding something deeply
  • the ability to understand is the ability to change perspectives
  • the ability to change perspectives requires the faculty of imagination
  • imagination is important in mathematics
  • patterns = mental models

I feel like I have been writing about the same things year after year. While this may be evidence that my intellectual growth has been stagnant, I am always happy to go back to the fundamentals and clarify what I really think (my mental models, so to speak), for myself more than for anyone else. Insha’Allah, I will continue to optimize my mental models so that they resemble more and more to the reality, and I become closer and closer to Truth.

Posted in Psychology, Thoughts | Tagged , , , , , | Leave a comment

My Reaction to the Continuum Hypothesis Story

Note: This was another short paper I wrote for my Foundation of Mathematics class back in March of 2015 (can’t believe it’s been 3 years!). I will try to publish some papers I wrote in the past on this blog so I can (1) have a more complete record of my thoughts, (2) better organize my thoughts, iA.

Update on life: passed CPA exam, thinking about marriage and looking for a job; miss math


11 March 2015

My Reaction to the Continuum Hypothesis Story

The continuum hypothesis was first proposed by Georg Cantor in 1878 and states that there does not exist an infinite set whose cardinality is between that of the integers and that of the reals. Ever since then, mathematicians had been trying to prove the hypothesis to be either true or false, but none succeeded. Due to the difficulty of the problem, Hilbert listed the continuum hypothesis problem as the first of the 23 problems he presented in 1900. In 1940, Godel proved that in any axiom system, there are statements that are not provable, and these statements are termed “undecidable”; he also found that the the continuum hypothesis cannot be disproved in the axiom system of set theory. In 1963, Paul Cohen proved that it cannot be proved, either. And with this proof, we finally have an answer to Hilbert’s first problem: continuum hypothesis is an undecidable mathematical statement in set theory.

My immediate emotional reaction to the story was fascination and slight unease. I was fascinated at the possibility that a mathematical statement could be undecidable, but somehow uneasy because I felt that the perfect world of mathematical objectivity in my mind was being harmed in some way. Fortunately, my slower intellectual reaction has provided some relief to my unease (but only elevated my fascination). This intellectual reaction will be the topic of this paper.

In order for my reaction to make sense, though, I will first have to describe my deeply held view of mathematics. I have always thought of mathematics as having its own existence independent of our minds. I cannot imagine, for example, that the statement 1+1=2 is false just because human beings don’t exist. To me, mathematical entities seem to exist in an eternal and unchangeable state (in a platonic mathematical world, to use Roger Penrose’s phrase), and only some mathematical entities and some statements about their relationships to each other have been discovered by our mental efforts so far. As a consequence of this belief, I think of the mathematical world as having a more constant reality than the mental world, meaning that a human thought may or may not have a truth value, but a mathematical statement definitely has to be either true or false because they are about entities that are constant, eternal, and unchangeable. To be clear, what I mean by a mathematical truth is what corresponds to the mathematical reality.

When I learned that there are undecidable mathematical statements, my first intellectual reaction was to try to reconcile this fact with my existing view of mathematics as described above. In my view, it is necessary for every mathematical statement to be either true or false. Does this contradict with Godel’s incompleteness theorems? The answer seems to be no, for even if a statement is not provable, meaning that there is no way to logically show it to be true or false, it does not necessarily follow that the statement does not have a truth value. There seems to be no contradiction here. If, however, in addition to the aforementioned platonic view of mathematics, one also holds the belief that every mathematical statement can be accessed or discovered (to use the term I used before) by our mental effort, then one will perhaps find a contradiction here.

Personally, I am still not sure whether I hold this latter belief, for I do not have a very deep understanding of how human intelligence is able to access mathematical knowledge (this seems to a question that lies in the intersection of epistemology and philosophy of mathematics). But suppose that I do hold this belief—how then can I reconcile my platonic view of mathematics (every mathematical statement has to be either true or false) with the fact that there are undecidable statements such as the continuum hypothesis?

One possibility is to adopt classical modal logic and classify the statements in the mathematical world as either necessarily, possibly, or contingently true or false[1]. Furthermore, we should also apply these notions to the physical world to make things even more clear. In this way, we will have a system in which even undecidable statements can be classified as true or false.

To see how this works, consider the parallel postulate. If we accept the parallel postulate as an axiom, then we have Euclidean geometry. If we don’t accept the parallel postulate as an axiom, then we have a non-Euclidean geometry like hyperbolic geometry. So the key question here is: is the parallel postulate itself true or false? According to the modal logical framework I presented above, the statements “If parallel postulate, then Euclidean geometry.” and “If not parallel postulate, then non-Euclidean geometry.” would both be necessarily true in the mathematical world, but the parallel postulate itself wouldn’t be necessarily true, but only contingently true, since its truth value depends on the axiom system you choose to adopt. In the physical world, however, it would appear that the parallel postulate has to be either necessarily true or necessarily false, depending on which geometry (Euclidean or one of the various non-Euclidean geometries) is the one that actually describes the universe. So, if it turns out that it is hyperbolic geometry that is actually the geometry of the universe, and not Euclidean geometry, then the parallel postulate will be necessarily false in the physical world.

Similarly, the statements “If we accept the continuum hypothesis, then we have set theory A.” and “If we don’t accept the continuum hypothesis, then we have set theory B.” are both necessarily true in the mathematical world. However, the continuum hypothesis itself would be only contingently true in the mathematical world, since its truth value again depends on which axiom system (set theory A or B) you choose to adopt. As for the truth value of the continuum hypothesis in the physical world, the answer would depend, for one thing, on whether one believes that physical (e.g. temporal, spatial) infinities are possible. Then there is also the question of what would be the physical equivalent of the set of integers? The set of reals? Perhaps string theorists who study the multiverse can better answer this question.

So, in this framework, the fact that the continuum hypothesis is undecidable poses no threat to my view that every mathematical statement has to be either true or false even if I believe that all mathematical knowledge is accessible by the human mind.

Another possibility is that the continuum hypothesis and other undecidable statements do have definitive truth values, we just haven’t developed the mathematical technology (e.g. new rules of proof; a new kind of logic etc.) to be able to show that. Of course, this is merely speculation, but I can see the possibility that the only thing that prevents us from determining whether the continuum hypothesis is true or not is just that we aren’t seeing the big picture. So, just as in the story of the blind men and an elephant, we are currently “blind” to the totality of truth concerning set theory. Another analogy is how we can only see 3D representations of a spinning tesseract, but never see the tesseract directly in its four-dimensional shape. In this analogy, the tesseract is “truth about set theory” and the 3D representations of different aspects of the tesseract are all the different possible axiomizations we could have of set theory. Hopefully, if we continue to examine all the different 3D representations of the tesseractset (different set theories) and how one representation changes to another (what makes one set theory different from the other; the overall logic etc.), then perhaps we will have the chance to really “see” the tesseract (the totality of truth concerning set theory) directly. When that happens, we may be able to see the exact status of the continuum hypothesis in the mathematical world. But, needless to say, all this is wildly speculative.

Yet another possibility is that the continuum hypothesis is a badly posed problem based on insufficient knowledge or the same issue discussed in the last paragrah. But I don’t know nearly enough about what qualifies a problem as “well-posed” or “badly-posed” to elaborate on this possibility.

To conclude, although the continuum hypothesis story has made me feel slightly uneasy, it has challenged my view of mathematics and forced me to find ways to modify my view to allow the possibility of undecidable statements. In this process, the continuum hypothesis story has enriched my intellectual life and launched me on a quest to develop an ever more accurate picture of the mathematical world.

[1]According to the Wikipedia Article “Modal Logic”: In classical modal logic, a proposition is said to be possible if and only if it is not necessarily false (regardless of whether it is actually true or actually false); necessary if and only if it is not possibly false; and contingent if and only if it is not necessarily false and not necessarily true (i.e. possible but not necessarily true).

Posted in Mathematics, Philosophy, Uncategorized | 1 Comment

Anatomy of an Illness

It has always been my goal to write at least a little bit about what I read so I remember the most important things I learned from my reading. However, I have failed 99% of the time. It’s the Easter holiday and I am in the rare situation where I have the time, energy, and motivation to write on this blog, and I will do so now.

Today, I finished reading Anatomy of an Illness as Perceived by the Patient by Norman Cousins. The book was published in 1979. It’s a short book about how the mind can help one’s body heal itself. The stories are inspirational and empowering. I am amazed again at the life force as a natural phenomenon.

What struck me the most about this book, however, is how little America has changed since 1979. Americans are still addicted to painkillers. Actually, I think it’s worse now than in 1979, with opioid addiction being such a big problem. Doctors still haven’t started providing alternative ways to pain management, and people still don’t understand that “pain is not the enemy”. As the story on leprosy and Drs. Paul & Margaret Brand so graphically demonstrates, pain is a blessing. Without pain, we cannot recognize that our body is being damaged. The best thing to do when there is pain to treat the underlying cause. Taking painkillers should be the last resort because not only do they not treat the underlying cause of the pain but they also damage the body in other ways (e.g. taking even one aspirin causes internal bleeding).

The rest of this post will be in the form of quotes from the book followed by my comments. I have discovered that this is a very easy way to write about what I read, because all I have to do is flip through the book, type up the sentences I underlined, and write my thoughts on those sentences. The downside of this method is that the post is less readable and writing in such a way does not aid me with the organization of my thoughts (which is often why I write in the first place).


If negative emotions produce negative chemical changes in the body, wouldn’t the positive emotions produce positive chemical changes? Is it possible that love, hope, faith, laughter, confidence, and the will to live have therapeutic value? (p.34-35)

Somehow, just reading this made me feel a small surge of positive emotions. It feels amazing simply to contemplate on the power of love, hope, and faith, etc. These intangible things are what make us human after all.


Studies show that up to 90 percent of patients who reach out for medical help are suffering from self-limiting disorders well within the range of the body’s own healing powers. (p.55)

That’s cool. The problem is, how do we activate our body’s own healing powers without relying on outside sources like placebo and the doctor’s authority?


The placebo is proof that there is no real separation between mind and body. Illness is always an interaction between both. (p.56)

This quote summarizes the central idea of the book.


It used to be assumed that there was some correlation between high suggestibility and low intelligence, and that people with low IQs were therefore apt to be better placebo subjects. This theory was exploded by Dr. H. Gold at the Cornell Conference on Therapy in 1946. The higher the intelligence, said Dr. Gold on the basis of his extended studies, the greater the potential benefit from the use of placebos. (p.63)

This is consistent with what I read in the Wikipedia article on hypnosis, i.e. high suggestibility is correlated with high intelligence. I find that highly intriguing. As mentioned in Thinking, Fast and Slow, another trait that correlates with high intelligence is high self-control. It’s a lot to think about. Honestly, I find human will to be even more fascinating than human intelligence.


Our experiences come at us in such profusion and from so many different directions that they are never really sorted out, much less absorbed. The result is clutter and confusion. We gorge the senses and starve the sensitivities. (p.65)

This quote is about stress produced by the modern lifestyle. I totally agree with this characterization. Indeed, we gorge the senses and starve the sensitivities. Who even talk about sensibilities anymore? I don’t remember exactly how he says it, but in the introduction of Critical Path, Buckminster Fuller talks about how poets are different from other people because they are really good at “feelings”. That has stuck with me because I constantly notice that modern life is destroying my ability to have delicate feelings. Most of the times, I just feel numbed. I am almost encouraged to feel numbed to get through the day. Sometimes, I feel like a robot, working through a long, long to-do list. That’s not the way to live, not if you want to live a human life.


In the end, the greatest value of the placebo is what it can tell us about life. … The placebo is only a tangible object made essential in an age that feels uncomfortable with intangibles, an age that prefers to think that every inner effect must have an outer cause. … If we can liberate ourselves from tangibles, we can connect hope and the will to live directly to the ability of the body to meet great threats and challenges. (p.66-67)

Ours is a materialistic age. We recognize the power of the mind but cannot incorporate that into our scientific worldview because our science is not advanced enough. It’s a shame, but I guess we’ll just have to wait till we have scientific explanations for consciousness for a true paradigm shift. Until then, we will likely stay in the materialistic age, our potentially powerful minds remain weak and trapped by materials.


“The answer to helplessness is not so very complicated.” Don Pablo said. “A man can do something for peace without having to jump into politics. Each man has inside him a basic decency and goodness. If he listens to it and acts on it, he is giving a great deal of what it is the world needs most. It is not complicated but it takes courage. It takes courage for a man to listen to his own goodness and act on it. Do we dare to be ourselves? This is the question that counts.” (p.79)

I think this is a beautiful quote by Pablo Casals. It reminds me of a philosophy lecture on “authenticity” I listened to a long time ago. If we all dare to be ourselves, the world would probably be better place. Of course, an individual is not just one thing. An individual is not all good or all bad, but I think in most of us, the good outweighs the bad. We just need more courage to act on that goodness within us…

I can relate to this. There is a voice inside me that tells me to pursue my dream despite all odds, to seek the truth and not waste anymore time seeking money. However, I consistently ignore that voice out of fear. I fear poverty and I fear being viewed as strange and unsuccessful by others. The result is that I am now going to be an accountant instead of a philosopher. I don’t have the courage to listen to the goodness inside me.


The most important thing about science is the scientific method–a way of thinking systematically, a way of assembling evidence and appraising it, a way of conducting experiments so as to predict accurately what will happen under given circumstances, a way of ascertaining and recognizing one’s own errors, a way of finding the fallacies in long-held ideas. (p.120-121)

This sentence on the scientific method reminds me of Popper and the central place of falsification in the scientific method. I think the scientific method is an extremely important and efficient way to discover truths about the physical world. However, in order to discover truths about the whole world (not just the physical), we need to supplement the scientific method with other truth-seeking methods. Maybe we could make a little room for rationalism in our quest for the truth. The world needs not only scientists, but also philosophers, theologians, and pure mathematicians…


Most doctors recognize that medicine is just as much an art as it is a science and that the most important knowledge in medicine to be learned or taught is the way the human mind and body can summon innermost resources to meet extraordinary challenges. (p.159)

I have always wondered: What exactly is the distinction between art and science the way people use these terms? The author seems to be suggesting here that art involves more human mind (and perhaps, “sensitivities”?) than does science.

Posted in Non-fiction, Psychology, Thoughts | Tagged , , , , , , , , , | Leave a comment

You Just Don’t Understand

A week ago, I finished reading You Just Don’t Understand: Women and Men in Conversation by sociolinguist Deborah Tannen. The central idea is that men and women have very different conversation styles resulting from different ways of viewing their relationship with others. Men tend to rank people in a hierarchy and view their relationship with others as an adversarial relationship, whereas women tend to see people as a community and view themselves as part of a network of connections. As a result, men tend to speak to establish status and demonstrate power and women tend to speak to establish connection and preserve intimacy. Also, men tend focus on the literal message being conveyed while women often focus on the meta-message behind the literal message. Another juxtaposition of the two styles is “report-talks” vs. “rapport-talks”. The book also covers many other topics, but those points are what I remember a week after finishing the book.

Reading the book was enlightening and opened my eyes to a whole new world that exists parallel to mine. In particular, it helped me understand why I can never have prolonged conversations with my dad, why he gets angry when I talk about my problems, why my younger brothers disrespect me and always resist my authority as the big sister, why a lot of men tend to speak only about themselves and doesn’t ask me questions, etc. I felt like I was learning about the culture of a foreign country.

Although the author tries to provide a balanced view and repeatedly points out that one conversation style is not better than the other but simply different, I feel strongly that women’s conversation style is superior to that of men’s, and I get the impression that deep down, the author feels the same way. I feel that women’s conversation style is better for the world because it is rooted in a desire for harmony and peace. Competition is good, but only in a context of a peaceful community.

Of course, there is downside to women’s conversation style. Talking about other people and sharing problems is how some women connect to each other and preserve intimacy. However, when you start talking badly about others (i.e. gossiping) and talk about your problems excessively and pressure others to share similar problems, then that’s too much and becomes a negative thing. We need to beware of this.

Also, in a world dominated by men, women are at a disadvantage because the way women speak reinforce men’s belief that women are indecisive, insecure, and weak. Women may be weaker than men physically, but we are not weaker than men emotionally and intellectually. Sadly, the way we speak make some men think that women are weaker in everything. In order to be successful in a men’s world, it is sometimes necessary for women to speak more like men.

The way men speak is not all bad, of course. But throughout history, it has always been women who adapt to men’s conversation style. I think it is time for men to start learning from women’s conversation style.

Interesting quotes:

The most important point is that gender distinctions are built into language. The words available to us to describe women and men are not the same words. And, most damaging of all, through language, our images and attitudes are buttressed and shaped. Simply by understanding and using the words of our language, we all absorb and pass on different, asymmetrical assumptions about men and women.(p.243)

…when trying to negotiate mutual preferences and decisions, women are often more indirect than men. But when it comes to talking about their personal relationships and feelings, many men are more indirect. (p.276)

Goffman points out that men are to women as adults are to children: loving protectors who will hold open doors, offer the first portion of sweets, reach high shelves, and lift heavy loads. But along with the privileges of childhood come liabilities: Children’s activities are interruptible, their time and territory expendable. Along with the privilege of being protected comes the loss of rights, and not being respected and treated like a full-fledged person. (p.287)

“Born rebels” who defy authority are not oblivious of it, but oversensitive to it. Defying authority is a way of asserting themselves and refusing to accept the subordinate position. When they are old enough, or established enough, to take the dominant position, reinforcing authority becomes the way to assert themselves, since the hierarchy is now operating to their advantage. (p.291)

Posted in Non-fiction, Psychology | Tagged , , , , | Leave a comment