Marriage and the Code of Canon Law: a quandary about Christian disparity of cult

This post is going to be a bit of a mess, mostly because I’m an amateur clumsily stumbling my way through the code of canon law (and I only have access to the 1983 code of canon law, available here, which has been updated in various and important ways since its first edition). There is one part of canon law which, ever since I first discovered it when exploring the Catholic faith, has made me extremely uncomfortable. I’ve been struggling with it practically since having become a Catholic. It concerns the correct understanding of the (in)validity of certain marriages under the code of canon law of the Catholic Church, and how that should inform the moral attitudes of Catholics with regards to those relationships. What the code of canon law suggests, which I will explain, seems prima facie morally implausible, and has the potential to put faithful Catholics in tremendously emotionally difficult situations. What it implies, in short, is that marriages between baptized Catholics and baptized evangelicals for which no dispensation from the Catholic Church is obtained are, in fact, invalid. I will briefly explain and explore this problem below.

To get some basic principles on the table, let’s begin by noting that there is a difference between the practice or discipline of the Catholic Church, and the faith of the Catholic Church. For instance, having a celibate priesthood is a matter of discipline, and if it changed tomorrow not one iota of the theology of the Catholic Church would have changed. There have also been, at various times and in various ways, poorly thought out rules governing these kinds of disciplines and practices. However, it is contrary to faith to think that the disciplines of the Church are in no way connected to her faith. We must regard canon law, even when imperfect, as an indication of what the Church teaches and believes.[1]

Some basic distinctions to keep in mind, when thinking about what the Catholic Church believes about marriage, include the following: according to Catholicism, marriages can be valid or invalid, and they can be sacramental or non-sacramental. To be a valid marriage, whether secular or religious (whether Catholic or not), the marriage has to have certain properties. For example, if somebody gets married on false pretense then the marriage is not valid, so a marriage must be based on authentic consent (which cannot be given on false pretense). To illustrate this point, imagine if a woman loved her fiancé, but the man who showed up on the day they were set to be married was an imposter – it was actually her fiancé’s evil identical twin brother. If she made vows based on the understanding that the man before her was her fiancé, is she married to the fiancé, his twin brother, or neither? The Catholic Church teaches that a marriage failed to take place in such a situation because of a defect in the consent given, so that she remains, in reality, unmarried. The same can be said if her fiancé turned out to be a woman who was pretending to be a male, or if her fiancé was secretly a CIA operative only ‘marrying’ her for professional reasons, and so on. The marriage would also be invalid if the would-be husband is impotent, for marriage is impossible where the marital act is impossible. Although Canon 1107 specifies that: “Can.  1107 Even if a marriage was entered into invalidly by reason of an impediment or a defect of form, the consent given is presumed to persist until its revocation is established[,]”[2] presumption is not to be confused with validity.

The point is that for marriage to take place at all it has to involve legitimate vows. So, canon law states:

“Can.  1095 The following are incapable of contracting marriage:
1/ those who lack the sufficient use of reason;
2/ those who suffer from a grave defect of discretion of judgment concerning the essential matrimonial rights and duties mutually to behanded over and accepted;
3/ those who are not able to assume the essential obligations of marriage for causes of a psychic nature.
Can.  1096 §1. For matrimonial consent to exist, the contracting parties must be at least not ignorant that marriage is a permanent partnership between a man and a woman ordered to the procreation of offspring by means of some sexual cooperation.
§2. This ignorance is not presumed after puberty.”[3]


“Can.  1083 §1. A man before he has completed his sixteenth year of age and a woman before she has completed her fourteenth year of age cannot enter into a valid marriage.”[4]

Moreover, a marriage is considered sacramental by the Catholic Church if and only (i) it is a valid marriage, and (ii) both persons involved have received the sacrament of baptism (whether in the Catholic church or not). Therefore, from a Catholic perspective, all baptized protestants who marry each other are sacramentally married.

Now, sacramental marriage is special in several ways, and most notably in this respect: that it is indissoluble, for the Bible says “therefore what God has joined together, let no one separate” (Mark 10:9). That means that divorce is, for Christian marriages, a legal fiction, and an impossibility in the eyes of God (according to the Catholic faith). The vows of marriage (including ‘until death do us part’ and ‘for better or for worse’) are taken to be absolutely binding. The vows allow for no exceptions, and if people give those vows validly they are bound to them indissolubly.

Non-Christian marriages are a different story. Although Canon 1141 says “A marriage that is ratum et consummatum can be dissolved by no human power and by no cause, except death,”[5] the Church (having no mere human power) reserves the right to exercise the dissolution of marriage in two cases:

“Can.  1142 For a just cause, the Roman Pontiff can dissolve a non-consummated marriage between baptized persons or between a baptized party and a non-baptized party at the request of both parties or of one of them, even if the other party is unwilling.
Can.  1143 §1. A marriage entered into by two non-baptized persons is dissolved by means of the pauline privilege in favor of the faith of the party who has received baptism by the very fact that a new marriage is contracted by the same party, provided that the non-baptized party departs.”[6]

Canon 1143 shows that valid marriages which are not sacramental (say, a marriage between two Hindus or two atheists) can be dissolved, and aren’t considered indissoluble by their very nature. Canon 1142 technically seems to imply either that sacramentally married persons can get a divorce if they haven’t yet consummated the marriage, or else that the matter and form of the sacrament of marriage haven’t been satisfied without consummation. Obviously most Catholics would presume the latter, but the issue of whether consummation constitutes part of the matter or form of marriage has never been explicitly settled in Catholic doctrine. The Catholic Encyclopedia has an interesting paragraph on this:

“To complete our inquiry concerning the essence of the Sacrament of Marriage, its matter and form, and its minister, we have still to mention a theory that was defended by a few jurists of the Middle Ages and has been revived by Dr. Jos. Freisen (“Geschichte des canonischen Eherechts”, Tübingen, 1888). According to this marriage in the strict sense, and therefore marriage as a sacrament, is not accomplished until consummation of the marriage is added to the consent. It is the consummation, therefore, that constitutes the matter or the form. But as Freisen retracted this opinion which could not be harmonized with the Church’s definitions, it is no longer of actual interest. This view was derived from the fact that marriage, according to Christ’s command, is absolutely indissoluble. On the other hand, it is undeniably the teaching and practice of the Church that, in spite of mutual consent, marriage can be dissolved by religious profession or by the declaration of the pope; hence the conclusion seemed to be that there was no real marriage previous to the consummation, since admittedly neither religious profession nor papal declaration can afterwards effect a dissolution. The error lies in taking indissolubility in a sense that the Church has never held. In one case, it is true, according to earlier ecclesiastical law, the previous relation of mere espousal between man and woman became a lawful marriage (and therefore the Sacrament of Marriage), namely when a valid betrothal was followed by consummation. It was a legal presumption that in this case the betrothed parties wished to lessen the sinfulness of their action as much as possible, and therefore performed it with the intention of marriage and not of fornication. The efficient cause of the marriage contract, as well as of the sacrament, was even in this case the mutual intention of marriage, although expression was not given to it in the regular way. This legal presumption ceased on 5 Feb., 1892, by Decree of Leo XIII, as it had grown obsolete among the faithful and was no longer adapted to actual conditions.”[7]

The matter remains unsettled.

A common misunderstanding worth addressing, in passing, concerns the Church’s practice of giving annulments. The Catholic Church can ‘annul’ marriages by giving an annulment, but this is not to be confused with divorce, for an annulment simply removes the Church’s presumption of validity after an investigation has established, to the satisfaction of the bishop(s), that some impediment to marriage existed in the first place which prevented the marriage from ever having taken place in reality. Annulments are merely recognitions that so-called marriages were never valid in the first place. Often those seeking annulments cannot get them because there is no good reason to think that the marriage they want annulled was invalid.

Having laid out the basic principles, let me proceed to lay out the basic problem.

“Can.  1059 Even if only one party is Catholic, the marriage of Catholics is governed not only by divine law but also by canon law, without prejudice to the competence of civil authority concerning the merely civil effects of the same marriage.”[8]

This canon implies that Catholics, whether they are practising their faith or not, are, as a matter of fact, governed by canon law. A Catholic can marry a non-Catholic (whether Christian or not) so long as they get a dispensation to do so from the Church.

A necessary condition for a marriage to be sacramental is for it to be valid. Anybody who was baptized in the Catholic church and hasn’t formally defected from the Church by a formal act cannot be validly married to a non-Catholic without a dispensation from the Church (this is generally referred to as a dispensation for a disparity of cult (can. 1129)). In the absence of that dispensation, however, the marriage between a Catholic and a non-Catholic baptized Christian is not sacramental because it is not valid.

“Can.  1124 Without express permission of the competent authority, a marriage is prohibited between two baptized persons of whom one is baptized in the Catholic Church or received into it after baptism and has not defected from it by a formal act and the other of whom is enrolled in a Church or ecclesial community not in full communion with the Catholic Church.”[9]


“Can.  1117 The form established above must be observed if at least one of the parties contracting marriage was baptized in the Catholic Church or received into it and has not defected from it by a formal act, without prejudice to the prescripts of ⇒ can. 1127, §2.”[10]

And again:

“Can.  1086 §1. A marriage between two persons, one of whom has been baptized in the Catholic Church or received into it and has not defected from it by a formal act and the other of whom is not baptized, is invalid.
§2. A person is not to be dispensed from this impediment unless the conditions mentioned in cann. 1125 and 1126 have been fulfilled.”[11]

Those exceptions are:

“1/ the Catholic party is to declare that he or she is prepared to remove dangers of defecting from the faith and is to make a sincere promise to do all in his or her power so that all offspring are baptized and brought up in the Catholic Church;
2/ the other party is to be informed at an appropriate time about the promises which the Catholic party is to make, in such a way that it is certain that he or she is truly aware of the promise and obligation of the Catholic party;
3/ both parties are to be instructed about the purposes and essential properties of marriage which neither of the contracting parties is to exclude.
Can.  1126 It is for the conference of bishops to establish the method in which these declarations and promises, which are always required, must be made and to define the manner in which they are to be established in the external forum and the non-Catholic party informed about them.”[12]

This presents us with our first difficulty to be considered. Suppose, assuming everything canon law says, that a Catholic who isn’t practising their (Catholic) faith, but hasn’t formally defected from the faith, gets married to a baptized evangelical who is practising his/her evangelical faith. In such a case the marriage between that Catholic and their evangelical spouse is not valid. Indeed, since canon law implies that Catholics are all governed by canon law (in the eyes of God), such a couple would (presumably) be living in sin because sexual union would not, for them, be a consummation of marriage at all. Even if the non-practising Catholic had become a ‘born again’ evangelical, and intended to get married in order follow God’s will for their lives, they would be objectively living in grave sin insofar as they were living the married life. Strangely, this means that the non-Christian couple who couldn’t care less about their relationship in the eyes of God have a genuine marriage relationship in God’s eyes, but that the Catholic-evangelical couple who care immensely about God’s will are sinning merely by living ‘the married life.’ Surely, though, that seems strange and morally counter-intuitive.

In order to explain why this is so, a lot of work would have to be done (and digital ink spilt) to explain how, according to the Catholic faith, the (Catholic) Church herself is related to all Christians (as mater et magistraet imperium), and to explain what Christ’s fulfillment of the law implies about the Christian view of marriage. I have not the time or the energy to accomplish this task to my satisfaction here, but I will content myself with a few passing remarks. First, Christians take themselves to not be bound to the letter of the Old Testament laws, including the ten commandments, because they are obligated to fulfill that to which the laws directed man in the first place. Thus, when the Torah sets down laws engineered to avoid religious syncretism, Christians are to fulfill the law by avoiding syncretism with the evil in the world. When the Torah sets down laws which direct man to a certain view of marriage (as between one man and one woman, exclusively committed to each other with an inviolable love which reflects the love of Christ, which is tirelessly forgiving), Christians are obligated to fulfill the essence of that law. Moreover, Christians live in a covenant with God through the Church, and the Catholic Church does, in fact, have the right to govern her children. She does so by other means as well, such as when granting indulgences. Therefore, for one who truly absorbs the Catholic vision of what this thing we call ‘the Church’ is, it becomes obvious that the canon law of the Church is binding upon all of the Church’s members (though the canon laws only regulate the lives of those considered to be ‘in communion’ with the Bishop of Rome). That ineloquent digression will have to suffice for now.

Coming back to the matter at hand, it seems like the appropriate reaction for the Catholic to have to the situation where a non-practising Catholic is living together with his/her evangelical spouse as though married is to assume that they are entirely innocent of any mortal sin. After all, the non-practising Catholic may have such a superficial understanding of Catholicism (and canon law) that their conscience is entirely clear because they are altogether unaware of their moral obligation to obtain a dispensation from the Church. Although sex outside of marriage may be gravely immoral, the Catechism of the Catholic Church says:

“1857 For a sin to be mortal, three conditions must together be met: ” Mortal sin is sin whose object is grave matter and which is also committed with full knowledge and deliberate consent.”…

1859 Mortal sin requires full knowledge and complete consent. It presupposes knowledge of the sinful character of the act, of its opposition to God’s law. It also implies a consent sufficiently deliberate to be a personal choice. Feigned ignorance and hardness of heart do not diminish, but rather increase, the voluntary character of a sin.”[13]


“1860 Unintentional ignorance can diminish or even remove the imputability of a grave offense. But no one is deemed to be ignorant of the principles of the moral law, which are written in the conscience of every man. the promptings of feelings and passions can also diminish the voluntary and free character of the offense, as can external pressures or pathological disorders. Sin committed through malice, by deliberate choice of evil, is the gravest.”[14]

Therefore, neither party commits any mortal sin. Indeed, if both parties are invincibly ignorant then it may be that they do not even commit venial sin, for if “the ignorance is invincible, or the moral subject is not responsible for his erroneous judgment, the evil committed by the person cannot be imputed to him.”[15] This helps mollify me significantly, but it by no means entirely dissolves the awkward tension.

To make matters worse, it is not possible to excuse non-practising Catholics by arguing that their failure to attend confession at least once a year (Canon 989), or receive Eucharist at least once a year (Canon 920 §1), or any other similar failure by omission, constitutes leaving the Church by a formal act. It would have been emotionally convenient if we could have argued that non-practising Catholics who get married outside the Church without dispensation inevitably and invariably satisfy the condition of having defected from the Church by a formal act (in some way), but this cannot be done. First, the criteria for such a formal act has never been clearly defined by the Church (so canon law provides no clear direction on the matter). Second (and more pertinently for us today), canon law has been abrogated since 1983 (which is the code of canon law I’ve been quoting from so far). There was a motu proprio issued by Pope emeritus Benedict XVI in 2009 titled Omnium in Mentem in which the 1983 code of canon law underwent explicit revision on this very point:

“Since the sacraments are the same for the entire Church, the supreme authority of the Church alone is competent to approve or define what is required for their validity and to determine the rites to be observed in their celebration (cf. can. 841). All this is equally applicable to the form to be observed in the celebration of marriage, if at least one of the parties has been baptized in the Catholic Church (cf. cann. 11 and 1108)…

Experience, however, has shown that this new law gave rise to numerous pastoral problems. First, in individual cases the definition and practical configuration of such a formal act of separation from the Church has proved difficult to establish, from both a theological and a canonical standpoint. In addition, many difficulties have surfaced both in pastoral activity and the practice of tribunals. Indeed, the new law appeared, at least indirectly, to facilitate and even in some way to encourage apostasy in places where the Catholic faithful are not numerous or where unjust marriage laws discriminate between citizens on the basis of religion. The new law also made difficult the return of baptized persons who greatly desired to contract a new canonical marriage following the failure of a preceding marriage. Finally, among other things, many of these marriages in effect became, as far as the Church is concerned, “clandestine” marriages.

In light of the above, and after carefully considering the views of the Fathers of the Congregation for the Doctrine of the Faith and the Pontifical Council for Legislative Texts, as well as those of the Bishops’ Conferences consulted with regard to the pastoral advantage of retaining or abrogating this exception from the general norm of can. 11, it appeared necessary to eliminate this norm which had been introduced into the corpus of canon law now in force.”[16]

It continues:

“Therefore I decree that in the same Code the following words are to be eliminated: “and has not left it by a formal act” (can. 1117); “and has not left it by means of a formal act” (can. 1086 § 1); “and has not left it by a formal act” (can. 1124).”[17]

As the Catholic Apologist Jimmy Akin notes: “The Holy See… seems to have come to the conclusion that the formal defection law was not working as desired, and so it has now gotten rid of the whole thing.”[18] Thus, prior to 2009 any baptized Catholic who formally defected from the Church (whatever that involved) was not obliged to seek a dispensation to get married outside the Church. However, as canon law currently stands, there are no exceptions: all Catholics must seek a dispensation to get married outside the Church.

This leaves us with a problem which can be difficult to navigate through emotionally. What obligations do practising Catholics have to the people they know and love who intend to marry but who, perhaps through no fault of their own, are invincibly ignorant of the moral imperative to obtain a dispensation from the Church? If their marriage cannot be valid in the eyes of God without a dispensation from the Church, then do practising Catholics have an obligation to inform people in these circumstances that their marriage will not be valid? It appears that while ignorance can excuse the couple preparing to live the married life, ignorance cannot excuse the Catholic who is aware of canon law. The Catholic who is aware that the relationship will be objectively gravely sinful seems to have a moral duty to inform the couple of this fact. Informing the couple of this fact may do more harm than good, because it has the potential to destroy relationships and create seemingly irreparable rifts between family members, not to mention giving an appearance of legalism in place of love. Nevertheless, it is a widely accepted and fairly well established fact that Catholics are never to treat people as means to ends instead of ends in themselves, and this means that Catholics cannot lie in order to safeguard the peace of mind of those who are engaged in grave sin, or to safeguard any of the relationships which may become vulnerable. Does the Catholic have the right to remain silent on the matter (rather than lying)? This question leads to some further difficulties.

Under older codes of canon law Catholics would not even have been permitted to attend a matrimonial ceremony of this kind, but under current canon law it is up to the discretion of Catholics whether they attend. Although I’m not aware of any canon law(s) binding Catholics to this, there is a common assumption among Catholics that if one attends a presumptively invalid marriage they have an obligation to make it clear that their attendance does not constitute their condoning the relationship (since there is something about it which makes it morally problematic). Michelle Arnold, writing for, advises Catholics as follows:

  • “The Church does not explicitly forbid Catholics from attending presumptively invalid marriages. Catholics must use their own prudential judgment in making the decision, keeping in mind the necessity to uphold the Catholic understanding of the sanctity of marriage. To make such a judgment, you might ask yourself if you believe the couple is doing the best that they can to act honorably and according to the truth that they have. For example, you might decide to attend the presumptively invalid wedding of a couple who is expecting a child (thereby attempting to provide a family for that child); but you might decline to attend the presumptively invalid wedding of a couple you know to have engaged in adultery (thereby destroying previous marriages and families).

  • While there may be just reason to attend a particular wedding that will be presumptively invalid, I cannot recommend participating as a member of the wedding party in such weddings. There is a difference between attending as a non-participating guest and actively involving yourself in the wedding.”[19]

Obviously such advice doesn’t come with any binding authority, but it does reflect a sense for where Catholic laity who are faithful to the Church land on such issues. So, if a Catholic (aware of canon law) is invited to a wedding they know will be a celebration of an invalid marriage, they will either decide to attend, or decline the invitation. If they do not attend they will undoubtedly be asked why they won’t be attending, and lying is not a (morally legitimate) option. If they do attend there is a real danger that their attendance will be understood as an endorsement and celebration of the relationship – a relationship which they believe to be objectively gravely sinful. It seems as though they would be morally compelled to inform the couple that their attendance should not be considered to indicate an unqualified approval of the relationship, but informing the couple of this fact makes the Catholic’s relationship to them vulnerable, and has the potential to harm various other relationships as well. Either option puts the Catholic in a tremendously precarious position.

This is the source of my discomfort concerning this discipline of the Church. Would it not be better for the spiritual welfare of everyone involved if the Church simply presumptively gave all Catholics this dispensation with the exception of clergymen (including the defrocked) and those from whom the Bishops have reason for actively withholding the dispensation? It seems as though it would be better, for it would increase the number of sacramental marriages, and sacraments involve supernatural grace. However, I am not expert enough in these matters to presume to advise canon lawyers, much less the Catholic Church.

Where does this leave the Catholic who finds herself in the circumstance I’ve been imagining (i.e., of being aware of canon law, and invited to attend a wedding between a non-practising Catholic and an evangelical Christian, who have not obtained a dispensation from the Church for their marriage)? As things currently stand, and although it is difficult (and requires discernment and tact), it seems to me that the Catholic does have a duty to inform the would-be spouses that, according to the teaching of the Catholic Church, their marriage isn’t a valid marriage unless they obtain a dispensation. However, one needn’t go into all the details about what this means (as though the couple wants or needs a lesson in canon law and/or Catholic moral theology). It suffices to make the couple aware that, on a technicality, from a Catholic perspective, their marriage will be ‘imperfect’ in a way which will concern them if ever they came (back) to the Catholic faith. In the eyes of the Catholic Church, the marriage will be considered invalid, so that if the content of the Catholic Church’s faith is true then the marriage will not be either sacramental or valid.

[1] I recall a quote from St. Thomas Aquinas, but I couldn’t find it. In it I recall him saying something about how it is contrary to faith to believe that the Church ever acts frivolously, and that faith enjoins us to submit to her authority even in matters of discipline. It is bothering me so much that I can’t find it – please, anyone who knows where Aquinas says something like this, share it in the comments.






[7] Augustinus Lehmkuhl, “Sacrament of Marriage,” in The Catholic Encyclopedia Vol. 9. (New York: Robert Appleton Company, 1910), April 28, 2016,






[13] CCC 1857, 1859

[14] CCC 1860

[15] CCC 1793





Math, therefore God?

“Monsier! (a+bn)/n=x, donc Dieu existe; répondez!”[1]

Thus (allegedly) spoke the mathematician Leonard Euler when, at the invitation of Russian Empress Catherine the second, he confronted Denis Diderot in a (very short) debate on the existence of God. Diderot, who was not very good at math, was dumbstruck; he had absolutely no idea how to even begin responding to such an argument. In fact, he couldn’t even understand the argument, and Euler knew it! The court laughed him literally out of town (he promptly asked the Empress for leave to return to France). The formula, of course, is entirely meaningless, and may have been sleight of hand on Euler’s part (making his argument mathemagical rather than mathematical). Additionally, the anecdote has survived only in sparse notes (of dubious historical relevance) here and there with probably varying degrees of accuracy, so it is anyone’s guess what Euler actually meant. This amusing anecdote does, however, invite us to think about what arguments there could be, in principle, from mathematics for the existence of God.[2] Without offering much commentary on how promising these arguments are, I want to distinguish three viable (or, at least, viably viable) types of arguments which could be constructed.

The Argument from Mathematical Beauty

Although the formula Euler originally spouted off didn’t signify anything of mathematical (or philosophical) consequence, the beauty of Euler’s equation, eiπ + 1 = 0, gave rise to the apocryphal anecdote that Euler argued “eiπ + 1 = 0, therefore God exists.” There is (mathematicians tell us) a sublime mathematical beauty in this equation, and there is no obvious or intuitive reason why it is true. What is so special about this equation? One savvy commentator I ran across online put it so nicely I feel compelled to quote him:

“It’s a sort of unifying identity in mathematics, containing each of the fundamental operations (additive, multiplicative, exponential) and each of the fundamental constants (e, i, pi, 1, 0) combined in a theorem that united trigonometry, analysis, and algebra and geometry. It’s really an amazing identity, and the proofs for it are diverse and fascinating…”[3]

It has, thus, been called the origin of all mathematics. Keith Devlin is purported to have said:

“like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler’s equation reaches down into the very depths of existence.”[4]

Its elegance cries out for an explanation, but that explanation has proved so elusive that a desperate appeal to God begins to look almost reasonable, even to (some) mathematicians.

What should we make of this sort of argument? It seems on its face to be about as prima facie (in)admissible as any other argument from beauty. However, this argument may have more to recommend it than meets the eye. In particular, mathematical beauty has an uncanny predictive ability, at least in the sense that the more beautiful the mathematical formula, the more likely it is to describe the fundamental structure of the real world. Robin Collins has noted, for instance, that:

“To say that the beauty of the mathematical structure of nature is merely subjective, however, completely fails to account for the amazing success of the criterion of beauty in producing predictively accurate theories, such as Einstein’s general theory of relativity.”[5]

John Polkinghorne, in a lecture I recently had the pleasure of listening to (via podcast), said something similar though with less economy of words:

“It isn’t just [to satisfy] an aesthetic indulgence that theoretical physicists look for beautiful equations; it is because we have found, time and again, that they are the ones which actually do describe… a true aspect of the physical world in which we live. I suppose the greatest physicist I’ve known personally was Paul Dirac, (who held Newton’s old chair… in Cambridge for more than 30 years, who was one of the founding figures of quantum theory, [and] unquestionably the greatest British theoretical physicist of the twentieth century) and he made his great discoveries by a relentless and highly successful lifelong quest for mathematical beauty. Dirac once said ‘it is more important to have beauty in your equations than to have them fit experiment.’ Now he didn’t mean by that that it didn’t matter at the end of the day whether your equations fit the experiments (I know no physicist could possibly mean that), but what he meant was this: ok, you’ve got your new theory, and you use the solution and you find it doesn’t seem to fit what the experimentalist is telling you – now there’s no doubt that’s a setback, but it’s not absolutely necessarily fatal. Almost certainly, you will have solved the equations in some sort of approximation, and maybe you’ve just made the wrong approximation, or maybe the experiments are wrong (we have [known that] to happen even more than once in the history of physics – even in my lifetime I can think of a couple examples of that), so at least there’s some sort of residual hope that with a bit more work and a bit more luck you might have hit the jackpot after all. But, if your equations are ugly, there’s no hope. The whole 300-year history of theoretical physics is against you. Only beautiful equations really describe the fundamental structure of the world. Now that’s a very strange fact about the world… What I am saying to you is that some of the most beautiful (mathematical) patterns that our pure mathematical friends can think up in their studies just thinking abstractly… are found actually to occur, to be instantiated, in the structure of the world around us.”[6]

So mathematical beauty satisfies the empirical desideratum of predictive power in the sense that the more beautiful the mathematical expression, the more likely it is to describe reality.

Interestingly I think this kind of consideration can motivate a scientist (and perhaps even a die-hard empiricist, and/or a naturalist) to believe in the objectivity of aesthetic properties. In fact, unless they find a plausible evolutionary account for why our brains should be calibrated so as to recognize more beauty in the abstract mathematical equations which, it turns out, describe reality, than we find in other equations, there will be a residual mystery about the eerie coincidence of mathematical beauty and accurate mathematical descriptions of physics. An eerie coincidence the queerness of which can perhaps be mitigated by admitting the objectivity of aesthetic qualities.

However, the puzzling queerness of that eerie coincidence can only be (or can most plausibly be) ultimately alleviated if the universe is seen as the product of a (trans-)cosmic artist. If behind the fundamental structure of the universe there lies an intellect with aesthetic sensibilities (in some sense), then that would explain why the world showcases the mathematical-aesthetic qualities it does at the level of fundamental physics even when there is no (obvious?) reason why it should have. That, though, begins to look quite a lot like Theism.

The Argument from the Applicability of Mathematics

This segues into the next kind of argument from mathematics, which concerns the applicability of mathematics to accurate descriptions of the fundamental structure of the physical world. For the purposes of this argument beauty is entirely irrelevant. What is surprising, and in need of an explanation (according to this argument), is that the physical world would turn out to be describable in the language of mathematics (and here we are not simply referring to the basic truths of arithmetic, which are true across all logically possible worlds). William Lane Craig has become the most well-known proponent of this argument, and his articulation of it is relatively succinct.

“Philosophers and scientists have puzzled over what physicist Eugene Wigner called the uncanny effectiveness of mathematics. How is it that a mathematical theorist like Peter Higgs can sit down at his desk and by pouring over mathematical equations predict the existence of a fundamental particle which experimentalists thirty years later after investing millions of dollars and thousands of man-hours are finally able to detect? Mathematics is the language of nature. But, how is this to be explained? If mathematical objects are abstract entities causally isolated from the universe then the applicability of mathematics is, in the words of philosopher of mathematics Penelope Maddy, “a happy coincidence.” On the other hand, if mathematical objects are just useful fictions, how is it that nature is written in the language of these fictions? In his book, Dr. Rosenberg emphasizes that naturalism doesn’t tolerate cosmic coincidences. But the naturalist has no explanation of the uncanny applicability of mathematics to the physical world. By [contrast], the theist has a ready explanation. When God created the physical universe, he designed it on the mathematical structure he had in mind. We can summarize this argument as follows:

  1. If God did not exist, the applicability of mathematics would be a happy coincidence.
  2. The applicability of mathematics is not a happy coincidence.
  3. Therefore, God exists.”[7]

I am not sure of this argument’s philosophical quality, since it seems to me that it may be a metaphysically necessary truth that a logically possible world be amenable to mathematical description of some kind. For instance, it certainly seems true that whatever the geometry of space happens to be, there’s no necessary fact of the matter, but it also seems true that if the geometry of space isn’t Euclidean, it may be hyperbolic, or elliptic, (or maybe something else, je ne sais quoi) but it has got to be something, and what it happens to be may, therefore, not cry out for any more explanation than any other quaint contingent fact about the world.[8] However, maybe I’m mistaken about this; maybe the argument is, in fact, just as viable as other teleological or ‘fine-tuning’ arguments are.

Argument from Mathematical Truth

Finally, the third kind of argument I can think of would go something like this: mathematical truths, like all truths, have truth-makers. These truth-makers will have to be metaphysically necessary on pain of mathematical truths being contingent – but it seems obvious that mathematical truths are necessary truths, that they hold across all logically possible worlds. Now, Nominalism about mathematical objects is incompatible with the commitments we just outlined (unless one adopts Nominalism about modal properties as well), and so seems implausible (or, at least, less plausible than it otherwise would have been in virtue of this incompatibility). Platonism also, however, seems to be problematic. Between Platonism and Nominalism, there is a wide range of views including Divine Conceptualism (according to which mathematical objects exist as necessary thoughts in the necessary mind of God), Theistic Activism, Scholastic Realism[9] and many others besides. In fact, a variety (and perhaps a majority) of the accounts of abstract objects on offer today presuppose the existence of God in different ways.

This opens the way to at least two arguments we could construct for the existence of God. First, we could argue that one of these accounts in particular is most plausibly correct (such as Greg Welty’s Theistic Conceptual Realism),[10] and work our way up from there to the implication that God exists. Second, we could take the disjunction of all the accounts of abstract objects which require the existence of God and argue that (i) if any of them is correct then God exists, but (ii) it is more plausible than not that at least one of them is correct, from which it follows (iii) it is more plausible than not that God exists.

So, there we have it, three kinds of arguments from mathematics for the existence of God; a transcendental argument (from beauty), a teleological argument (from applicability), and an ontological argument (from necessity). Could there be others? Maybe, but I suspect that they will all end up falling into one or another (or maybe at least one) of the general categories I tried to outline here. I admit that I didn’t outline them as general categories very well, but that exercise will have to wait for another day when I have more time to blog to my heart’s content.

As a quick post scriptum; if Euler had any substantive argument in mind and wasn’t merely mocking Diderot for his lack of mathematical aptitude, which of these three kinds of arguments would he most likely have had in mind? It’s hard to say, of course, but my best guess is that if he had anything in mind at all, it would fall into the third category. He may have been thinking that the fact that mathematical and purely abstract (algebraic) ‘structural’ truths exist at all requires some explanation, and this explanation must be found in God. This is just a guess, and I make no apologies for it – I am happy to think that Euler was just teasing Diderot, but I am equally happy to entertain the thought that if Diderot had not immediately asked to leave (because of his embarrassment), Euler may have been able to elucidate his point.

[1] Gillings, Richard J. “The so-called Euler-Diderot incident.” The American Mathematical Monthly 61, no. 2 (1954): 77-80.

[2] Notice that these are not to be confused with mathematical arguments per se; they are merely arguments from mathematics, in the same way as you might have arguments from physics (the argument from cosmological fine-tuning, the Kalam, etc.) for the existence of God which are not intended to be scientific proofs of God’s existence, but scientifically informed philosophical proofs/arguments for God’s existence.

[3] Russel James, Why was Euler’s Identity Supposed to be a Proof for the Existence of God,; Note that he finishes the quoted paragraph with the words “but It has nothing to do with god whatsoever.” I have left this out not because I think he is wrong, or to misrepresent his position, but because it has nothing to do with the formula and everything to do with the propositional attitude he adopts with respect to the question of whether the formula is any kind of reason to think there is a being like God.

[4] Paul J. Nahin, Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills, (Princeton University Press, 2011), 1.

[5] Robin Collins, The Case for Cosmic Design, (2008),

[6] John Polkinghorne, Science in the Public Sphere,

[7] William Lane Craig, Is Faith in God Reasonable? William Lane Craig vs. Dr. Rosenberg,

[8] I am really, honestly, no more sure of this counter-argument than I am of the argument. For those interested, please do check out the debate between Craig and Daniel Came on the Unbelievable? Podcast, which you can also find here:

[9] J.T. Bridges defends this view:

[10] See: Greg Welty, “Theistic Conceptual Realism,” in Beyond the Control of God: Six views on the Problem of God and Abstract Objects, ed. Paul Gould, (New York: Bloomsbury Academic, 2014), 81-96.

Is Presentism true(-making apt)?

Here’s an argument against presentism from the intuitively plausible principle that for any truth, there must be a truth-maker which makes a difference. Although I’m assuming a correspondence theory of truth, I don’t mind flirting with the idea that other theories of truth could adopt some truth-making principle and carry on with my argument keeping everything else the same. What I mean, here, by making a difference is something like this: if there is a possible world W1, in which T1 is true, and M1 is T1‘s truthmaker, then there is no possible world W2 which is maximally close to W1 in all respects save for that M1 (or its equivalent M2)[1] is unavailable (so, similar mutatis mutandis), at which T1 (or its equivalent T2) is true. I will bracket concerns about whether truths are multiply-realizable in the sense that any particular truth Tn might have any of a set of realizers {M1, M2, M3… Mn}, and I will, therefore, dodge questions about over-determinated truth-values and related concerns; I only note in passing that I don’t think much of these concerns, but I want to avoid them because I also haven’t thought much about these concerns.

Suffice it to say that by T1 having a truth-maker in M1, I mean that M1 is the reason T1 is true.[2] Now, consider the world as it looks through the eyes of a presentist. The presentist believes that the set of all things which exist, and the set of all things which exists now, are identical. There is no thing which both exists, and does not exist presently. Only the sum of all truths which are true ‘right now’ are true at all (this may be thought to be an unfair characterization, since the presentist may still believe that necessary truths have timeless entities, such as abstract objects, as their truth-makers, but I think this is contrary to the letter, if not the spirit, of presentism; for the strict presentist, only that which is present exists).

Now, what, on presentism, can account for the fact that we can make true statements about the past (let alone the future)? There is, presumably, some truth-making ingredient which presently exists which can make true statements about the past true. However, consider the classical problem in epistemology of the unverifiability of the reality of the past. As Bertrand Russell put it in his famous Problems of Philosophy, our knowledge of the past is based not on sense-experience (which would make it empirical in the strictest sense), but on our acquaintance with our own memory. We must, he insists, be able to have knowledge by acquaintance with things other than sense-experience, or else we would not be able to know that the past was real:

“But if [sense-data] were the sole example [of the things with which we are acquainted], our knowledge would be very much more restricted than it is. We should only know what is now present to our senses: we could not know anything about the past–not even that there was a past…
This immediate knowledge by memory is the source of all our knowledge concerning the past: without it, there could be no knowledge of the past by inference, since we should never know that there was anything past to be inferred.”[3]

The point here is that skepticism about the reliability of our memory will lead to skepticism about the very reality of the past, a worry which no amount of empirical investigation can alleviate. Suppose, for the sake of argument, that the whole world popped into existence moments ago with the appearance of age (e.g., light travelling to your retinas with the appearance of having come from stars (light-years away), memories of the first half of the conversation you may now be in, memories of having read the opening paragraph of this post, etc.). How would the world look any different from how it would have looked if it had had a past (and, let’s assume, precisely that past in which the evidence leads us to believe)? There seems to be no difference between the two. The worlds look empirically identical.

You might (quite rightly) think that these possible worlds could be metaphysically differentiated, but how could we articulate the metaphysical distinction on presentism? We cannot just say that only one of the worlds has the property of having a past, for whatever that amounts to on presentism will, it seems, be metaphysically indistinguishable from the property of having the appearance of having a past! Where, on the present ‘slice’ of the universe (whatever shape that takes), cosmos, and/or noumenal world can we locate a truth-maker for past-tense truths which would not have been there if the world had merely popped into existence moments ago preloaded with all the appearances of age? If there is no way to articulate a difference, then we might have on hand a good reason to be skeptical of presentism (or else, I suppose, skeptical of truth-making accounts of truth, but I take it as nearly incontestable that presentism is less intuitively secure than the generic truth-making account of truth). Maybe I’m mistaken, but I am under the spell of a powerful suspicion that the metaphysics of presentism makes no room for the kind of truth-maker I’m looking for.

But: perhaps there is a problem with the question itself. The presentist might insist that they take issue with the grammar of this objection, since it seems as though it assumes the reality of the past. The presentist may insist that on presentism there is no metaphysical difference between the possible world ‘with a real past’ and the possible world ‘with an apparent past’ precisely because there is no difference (i.e., simpliciter). The question becomes a pseudo-question, and the presentist follows the dance-moves of the positivist around the issue. This, however, seems to me to be an at least equally damning flaw in presentism as the lack of a truth-maker would be; if the presentist cannot make room for the meaningfulness of a distinction we all know very well to be meaningful then we should treat presentism with the same[4] disdain with which we treat logical positivism. If they do not make room for the semantic difference then they will either be violating the law of excluded middle, or they will be denying the meaningfulness of propositions we all know to be perfectly intelligible.

[1] I’m not sure if this comment, along with the one shortly to follow, is entirely appropriate here. I suspect that it may depend on one’s theory of reference across possible worlds. Nevertheless, I think my meaning is clear enough for the purposes of this post.

[2] Depending on how we cash out ‘reason’ here, this might be enough to avoid the difficulties I gestured towards above.

[3] Bertrand Russell, The Problems of Philosophy, (OUP: Oxford, 2001), 21.

[4] If not the same, on account of logical positivism’s being self-referentially incoherent (a flaw which we have yet to show presentism to have earned for itself), then at least a similar disdain. One might even say, a disdain so similar that the difference cannot be verified.

Some Considerations in Favor of Moral Realism

I was pulled into a facebook discussion today about moral realism. I decided I should use that as an excuse to write a short blog-post outlining some philosophical considerations which, I think, should lead us to affirm moral realism with confidence.

First, if there are properly basic beliefs which are not analytic, it seems that the belief in moral realism will definitely be properly basic. If we adopt a purely pragmatic account of epistemic justification then it also seems as though moral realism will be preferable to its negation. In fact, on a variety of epistemologies it looks as though moral realism fares pretty well. What kind of epistemology would justify moral anti-realism? I think the epistemologies which come to mind seem more philosophically suspicious than their competitors. So, unless we have a defeater for moral realism, we seem to be well within our rights to accept moral realism.

What might that defeater be? I suppose one could argue that if Naturalism is true, the moral anti-realism is true, but Naturalism is true, therefore et cetera. However, what reason do we have for believing that Naturalism is true? In all my time as a philosopher I have still yet to hear an even half-way decent argument for Naturalism. I would invite Naturalists to offer arguments here, but experience and my gut both tell me that most Naturalists have, at best, a vague sense that Naturalism seems right, and a poorly thought out set of reasons for thinking that metaphysical naturalism is true. Nevertheless, I remain open to incoming arguments, should anyone wish to present them. I should note, however, that even on Naturalism one should do whatever they can to make room for moral realism, for instance by trying to work out an account of Moral Naturalism.[1]

Second, there is a reductio ad absurdam we can run against arguments for moral anti-realism, which, if I recall correctly, W.L. Craig has presented. The idea goes like this: any argument you could give against moral realism can be parodied with near perfect parity into an argument against belief in the noumenal external world apprehended through the empirical senses. In the case of the external world we apprehend through the five senses that there is such a thing, which seems mind-independent and experience-independent. We have, of course, never verified that the external world is there absent any experience at all, and this is why we occasionally run into philosophers who adopt subjective idealism and deny that there is any such thing as a mind/experience-independent world. We are in a similar position with respect to our meta-ethical beliefs. In our moral experience we apprehend (through experience) that there are moral duties, values and facts which appear to be as objective as anything else we apprehend by experience. We naturally conclude that we encounter, in and through our moral experiences, a moral reality, a world of objective moral facts. In fact, our belief in moral realism is closer in kind to our belief in the noumenal external world than are our beliefs in mathematical facts or modal facts. The latter are the result of the operations of pure reason, whereas the former are the deliverances of experience.

There is, we think, something objectively real about the rock we touch, but this judgment is as much an intellectual knee-jerk reaction as it is possible to conceive. We have the same kind of intellectual knee-jerk reaction when it comes to moral realism, and the reaction comes with just as much force. It takes equally extreme cunning to convince ourselves to believe in moral nihilism as it does to fool ourselves into accepting subjective idealism. Both are just forms of skepticism. In fact, if one puts the arguments down on paper and compares them it will become obvious that there is no reason to deny moral realism which won’t count as an equally good reason to deny the noumenal external world. The subjective idealist won’t be impressed with this reductio, but most people will be.

Third, we can argue in the spirit and fashion of G.E. Moore, whose famous response to the skeptic was “I have a hand!” G.E. Moore’s point was that he would always be more sure that he had a hand than he could be that any argument for skepticism was sound. He might think that all the premises seem true, and agree that the argument seems logically valid, but he would deny that this gives him good enough reason to think that such an argument is sound. The credence which an apparently sound argument for skepticism provides would always, according to Moore, be outweighed by the credence given by experience for the proposition that he has a hand. No argument for skepticism, however good, will justify embracing skepticism, because no argument can make skepticism more plausible than things like ‘that I have a hand.’

To illustrate this point with an analogy, let’s use a logical argument for being skeptical of logical entailment. Suppose a teacher tells her elementary students that they will have a surprise quiz next week. Susie, a young student and budding logician, figures that the quiz wouldn’t be a surprise if it were on Friday, since they would have gone all week without it and would, therefore, be expecting it on Friday. She concludes that the surprise quiz can’t take place on Friday. However, she reasons that since Friday has been logically eliminated, the quiz cannot take place on Thursday either, since, if it hadn’t occurred until Thursday, but the quiz can’t possibly be a surprise Friday, then it can’t be a surprise Thursday either. She continues this process of elimination and determines that there is no day next week on which it is possible to have a surprise quiz. She has not made any obvious logical error in her reasoning, and yet just imagine her surprise when she has a quiz on Tuesday! Therefore, logical reasoning doesn’t always lead from true premises to true conclusions, even if it starts off with true premises and at each step the logical structure of the argument is impeccable.

What are we to make of such an argument? Well, we could come up with very clever responses, but Moore’s point is that even if we weren’t clever enough to discern where the reasoning is going wrong, we would (and should) still not accept that the argument justifies skepticism about logic! I am, and always will be, more sure of modus ponens than I can be that an argument for logical contradiction is sound. This is Moore’s point, and it translates well to the issue of moral realism/anti-realism.

The Atheist philosophy Louise Antony put it nicely when she said “Any argument for moral scepticism will be based upon premises which are less obvious than the existence of objective moral values themselves.”[2] If she is right then it seems like the Moorean response works as well here as it does anywhere.


One popular objection to the second argument I presented, which I hear surprisingly often, is that in our empirical experiences we can achieve a reasonable consensus about what the physical world is really like, but our moral experiences are less conspicuous, less clear, ‘fuzzier,’ and are less conducive to creating consensus about the fabric and structure of moral reality. This, it is suggested, gives us at least one reason to think that we should place more confidence in our empirical experience of the noumenal external world than we should place in our moral experience. Our moral experiences are more suspicious because they are a great deal vaguer than our empirical experiences.

I have two responses to this. First, although not all scientific matters can be adjudicated by empirical experiments either (think, for instance, of empirically equivalent scientific theories in equally good scientific standing, such as the neo-Lorentzian view of relativity, and the standard view of relativity), I will grant (and not just for the sake of argument) that scientific consensus is more easily reached than moral/ethical consensus. However, I want to note in passing that nearly everyone (who is some kind of moral realist) agrees that it is wrong to (without a justifying reason) kick a pregnant woman in the stomach repeatedly for amusement. There is, I think, a great deal more moral consensus than people typically imagine, but I digress. Having admitted this point of disanalogy, it still looks to me as though moral realism is so nearly as well justified as belief in the noumenal external world that the disanalogy makes no practical difference; moral realism ought still to be believed in the absence of a defeater, or else, on pain of inconsistency, we will be putting our belief in the external world in the very near occasion of philosophical abandonment.

My second response is to say that I think this objection confuses moral realism with particular meta-ethical accounts, or normative accounts. The arguments presented are not suggesting that we should be able to discover, through moral experience, what is objectively morally right (or wrong) with as much clarity and consensus as we discover what is objectively scientifically right (or wrong). What is being claimed is merely that in our moral experience we are as sure that we are being confronted with some set of objective moral facts as we are that, in our empirical experience, we are being confronted with an external world. The point here is that moral experience leads us to be confident not in any particular moral theory (eg. Utilitarianism, Egoism, Deontology, etc.), but in moral realism itself, a presumption which all moral theories share in common, and on which their coherence depends.


[1] See James Lenman, “Moral Naturalism,” inThe Stanford Encyclopedia of Philosophy ed. Edward N. Zalta, (2014).


Naturalism and Supernaturalism

What, exactly, is Naturalism? The naïve definition would go: Naturalism is the belief that there are no supernatural entities. What, though, are supernatural entities? The go-to example would be God, but that’s an example rather than a definition. As far as definitions go, a typical place to start is to say that a supernatural entity is anything which is empirically undetectable, or not verifiable/falsifiable by the scientific method. However, plenty of unquestionably scientific beliefs are in things which are not strictly falsifiable (such as the existence of our universe), and a ‘scientific’ view of the world often involves commitment to beliefs which aren’t strictly verifiable (such as the legitimacy of inductive reasoning, or the reality of the past). Moreover, this definition entails that moral values, the laws of logic, the fundamental principles of arithmetic (and all mathematics), aesthetic qualities, facts themselves (as model-independent truth-makers), propositions (whether necessary, contingent, or necessarily false), the (noumenal) external world, and even purely mental phenomena (eg. qualia), will all be supernatural. Science itself, it turns out, is replete with presumptions of supernaturalism according to the stipulated definition.

Alvin Plantinga once defined Naturalism as the belief that there is no such being as God, nor anything like God. I used to think that this definition was serviceable, but I have come to see that it invites some of the most egregious difficulties of all. Buddhists and Mormons may qualify as Naturalists on this definition, and mathematical Platonists may not qualify as Naturalists! Surely that can’t be right. A definition of naturalism on which it turns out that Joseph Smith is a naturalist and Frege a supernaturalist cannot be right. The notorious difficulty of defining Naturalism should now be evident. What once looked like a trivially easy task now appears to be a herculean feat; how are we to draw the line between the natural and the supernatural? To echo (mutatis mutandis) a famous saying of St. Augustine: if nobody asks me what Naturalism is, I know, but if you ask me, I do not know.

One could always suggest that the term ‘Naturalism’ has no definition precisely because concepts have no definitions. Wittgenstein’s famous suggestion that concepts like ‘GAME’ have no definition,[1] and Quine’s famous skepticism about analyticity,[2] are just two of many factors which have contributed to the recent retreat from ‘definitions’ in the philosophy of concepts.[3] This trend has led to the wide embrace of prototype theory, theory-theory, and other alternatives to the classical theory of concepts. If we must give up on definitions, it seems to me that we must largely give up on the project of analytic philosophy, and that makes me considerably uneasy; but then, I’ve always been squeamish about anti-rationalist sentiments. It may turn out we can do no better than to say something like that Naturalists adopt belief systems related by a mere family resemblance, but which cannot be neatly subsumed under one definition. I, however, (stubborn rationalist that I am) will not give up on definitions without a fight.

On the other hand, if Naturalism cannot be defined then those of us who wish to remain analytic philosophers can just cut our losses and accuse self-identifying naturalists of having an unintelligible worldview; one the expression of which involves a fundamental theoretical term for which no clear definition can be given. In other words, when somebody claims that Naturalism is true we can simply retort: “I don’t know what that means, and neither do you.” What kind of rejoinder could they give? Either they will provide us with an acceptable definition (so that we’ll have finally teased it out), or they will have to reconsider the philosophical foundations of everything they believe they believe. Win-win by my count.

In the meantime, let’s try on some definitions for size. Here’s one:

P is a naturalist =def. P is an atheist who believes that all that exists is discoverable by the scientific method.

This definition is bad for several reasons. To begin with, it isn’t clear that a Naturalist need be an atheist; why couldn’t they be a verificationist,[4] or a Wittgensteinian? It seems, at first blush, sufficient that one not believe that “God exists” is a metaphysical truth, but then it also seems wrong to say that an agnostic can be a naturalist. An agnostic is agnostic with respect to supernatural entities, but a naturalist is not. So we’re left in a quandary with respect to the first half of our definition.

The second half doesn’t fair much better. Apart from the fact that scientists routinely commit themselves to the reality of entities which are beyond the scope of strictly empirical discoverability (such as the existence of alternative space-times in a multiverse), there is an puzzle involved in stating what, precisely, qualifies as scientifically discoverable. For instance, many of the fundamental entities in particle physics are not directly empirically observable (they are, in fact, often referred to as ‘unobservable entities’), but we have good reasons to think they exist based on the hypothetico-deductive method (i.e., we know what empirical effects they would have if they did exist, and we can verify those). However, that amounts to having good scientific and empirical motivation for believing in unobservable entities. Is it impossible to have good scientific and empirical motivation for believing in ghosts, or numbers, or God? W.V.O. Quine famously stated that if he saw any empirically justifiable motivation for belief in things like God, or the soul, he would happily accept them into his ontology. In fact, in a move motivated by his commitment to his Naturalized Epistemology,[5] Quine did eventually come to accept the existence of certain abstract objects (namely, sets). Quine leaves us with two choices: either we say that even Quine wasn’t really a (metaphysical) Naturalist in the end, or we find a way to allow Naturalists to believe in things like numbers, moral values, aesthetic facts, and other things which we don’t usually think of as ‘Natural’ entities. I suggest we make use of the notion of scientific/empirical motivation; in other words, we should make room for Naturalists to work out an ontology motivated by a scientific view of the world. The only danger I foresee in that move is that if even belief in abstract objects can be scientifically motivated, it seems as though belief in God, or anything, might turn out to be possibly scientifically motivated. Nevertheless, let us consider a second definition:

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue, and that the only entities which exist are the entities to which the acceptance of a literal interpretation of science commits us.

The first half of this definition seems fine to me, so that’s some progress. The second half is problematic because it implies that constructive empiricists, for instance, are not naturalists; the constructive empiricist agrees with the scientific realist that the statements of science should be literally construed/interpreted, but that when we accept a scientific theory we commit ourselves only to (i) the observable entities posited by the theory, and (ii) the empirical adequacy of the theory. Since the constructive empiricist adopts an agnostic attitude towards unobservable entities, none of them would qualify as naturalists on the above definition. In fact, anyone who adopts any version of scientific anti-realism (including the model-dependent realism of Stephen Hawking and Leonard Mlodinow, or even structural realism) will be disqualified from the running for candidate naturalists.

Let’s try a third:

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue, and P believes in some of, and only, the entities to which a literal interpretation of science commits us.

A possible problem with this definition might be that it threatens to include solipsists (though it isn’t clear what in science, interpreted literally, would commit anyone to the existence of persons). Perhaps we should replace “entities to which the acceptance of a literal interpretation of science commits us” with something like “entities to which our best understanding of science commits us.” That might be problematic since what the best understanding of science is seems up for debate. Perhaps it should be changed to: “entities to which a legitimate interpretation of science commits us.”

P is a naturalist =def. P believes that “God exists,” interpreted as a metaphysical statement, is untrue; P believes in some of the entities to which a legitimate understanding of science commits us; P does not believe in any entities belief in which cannot be motivated by a scientific view of the world.

This definition isn’t obviously problematic. It looks to be about as good as I can do, off the top of my head. Note that if this definition is successful, then we have also found the definition of supernaturalism, since (obviously) the definition of naturalism and the definition of supernaturalism bear a symmetrical relation of dependence to one another. This still has some notable disadvantages, including that naturalists will not be able to justify believing in moral facts unless they can generate motivation for believing in them given the resources of a scientific worldview. However, those disadvantages may just come with the territory; they may be the disadvantages not of our definition, but of the philosophy of metaphysical naturalism.

One final note; the term ‘supernaturalism’ has a bit of a bad rep because it is popularly associated with things like ghosts, energies, auras, mind-reading, witchcraft and (for better or worse) a variety of religious beliefs. Because of this many philosophers have opted for using synonyms such as ‘ultra-mundane’ to refer to things like moral facts, possible worlds, necessary beings, et alia. I don’t much mind which term is used, but one advantage to retaining the use of the term ‘supernatural’ is that it helps focus our attempt to define ‘natural’ and its cognates. If we had to define the terms ‘natural’ and ‘ultra-mundane’ it might be less apparent that whatever qualifies as unnatural is going to qualify as ultra-mundane, and vice versa.

[1] Ludwig Wittgenstein, Philosophical Investigations second edition, transl. G.E.M. Anscombe (Blackwell Publishers, 1999).

[2] W.V.O. Quine, “Two Dogmas of Empiricism,” in The Philosophical Review vol. 60, no.1 (1951): 20-43.

[3] For more see: Stephen Laurence and Eric Margolis, “Concepts and Cognitive Science,” in Concepts: Core Readings (1999): 3-81.

[4] A verificationist, I mean, ‘about’ Theism.


Mlodinow’s Euclidean Equivocation

Leonard Mlodinow, who with Stephen Hawking co-authored The Grand Design, wrote the following passage in another book he wrote titled Euclid’s Window:

“By 1824, Gauss had apparently worked out an entire theory. On November 6 of that year, Gauss wrote to F. A. Taurinus, a lawyer who dabbled quite intelligently in mathematics, “The assumption that the sum of the three angles [of a triangle] is less than 180° leads to a special geometry, quite different from ours [Le., Euclidean], which is absolutely consistent, and which I have developed quite satisfactorily for myself. . ..” Gauss never published this, and insisted to Taurinus and others that they not make his discoveries public. Why? It wasn’t the church Gauss feared, it was its remnant, the secular philosophers.
In Gauss’s day, science and philosophy hadn’t completely separated. Physics wasn’t yet known as “physics” but “natural philosophy.” Scientific reasoning was no longer punishable by death, yet ideas arising from faith or simply intuition were often considered equally valid. One fad of the day which particularly amused Gauss was called “table-rapping,” in which a group of otherwise intelligent people would sit around a table with their hands placed in an arched position upon it. After a halfhour or so, the table, as if bored with them, would start to move or tum. This was supposedly some sort of psychic message from the dead. Exactly what message the ghouls were sending is unclear, although the obvious conclusion is that dead people like to position their tables against the far wall. In one instance, the entire Heidelberg law faculty followed for some time as their table moved across the room. One pictures a bunch of bearded, black-suited jurists pacing alongside, struggling to keep their hands in their appointed spot, attributing the locomotion to occult animal magnetism rather than their push. This, to Gauss’s world, was reasonable; the thought that Euclid had erred was not.”[1]

Ignoring the snide and historically fantastical notes about scientific reasoning ever having been punishable by death,[2] this passage is meant to set Gauss up as the champion of science, and his work is meant to signal the victory of science over philosophy. Mlodinow means to show how Euclid’s fifth postulate (concerning parallel lines), which was believed to be as philosophically secure as anything could be, was proven to be incorrect by the discovery, in modern physics, that space is non-Euclidean; in other words, it is not true that for any straight line L, and any point P not on L, there is only one line which can be drawn through P parallel to L. It may seem to be true, but our study of the physical world shows us that it isn’t (so the story goes). This, I will suggest, is just rhetorical slight of hand on Mlodinow’s part. In fact, it is worse; I believe that this represents a genuine antinomy in Mlodinow’s view of the nature of science itself, given what he has committed himself to in print elsewhere. I’ll develop this shortly below.

Shortly after the passage about Gauss trembling in fear of the indomitable secular philosophers (all of whom were apparently busy pushing tables around faculty lounges trying to communicate with the dead) Mlodinow introduces the character of Immanuel Kant. He chooses to portray Kant as the philosopher par excellence, and as the antagonist of scientific progress. He takes special care to note:

 “In Critique of Pure Reason, Kant calls Euclidean space “an inevitable necessity of thought.””[3]

It is worth saying a few words in defense of Kant, before we move on. Mlodinow provided no (precise) citation for the quote, so I searched through the Critique of Pure Reason for myself and could find no such statement. A quick glance at some Kant scholars indicates to me that there are some mixed signals here. On the one hand, never once does Kant refer to Euclid, or Euclidean geometry, or the fifth postulate, or even parallel lines (apart from one brief comment about immediately perceiving that the opposite angles of a parallelogram are equivalent) throughout the Critique of Pure Reason, so that the attribution to him of the saying above must, at best, be justified by reading between the lines. On the other hand, it is not unlikely that when Kant made statements about space, he was presupposing something like Euclidean space. It is true that Kant thought that the conception of space, like that of time, was an a priori intuition which was inalienable to the rational intellect. Instead of saying, with the empiricists, that our conception of space came come from the five senses, Kant thought that our conception of space was a precondition for our having intelligible empirical experiences at all. He writes:

“Space is not a conception which has been derived from outward experiences. For, in order that certain sensations may relate to something without me (that is, to something which occupies a different part of space from that in which I am); in like manner, in order that I may represent them not merely as without, of, and near to each other, but also in separate places, the representation of space must already exist as a foundation. Consequently, the representation of space cannot be borrowed from the relations of external phenomena through experience; but, on the contrary, this external experience is itself only possible through the said antecedent representation. Space then is a necessary representation a priori, which serves for the foundation of all external intuitions.”[4]

Although Kant’s student, Schultz, was apparently interested in the treatment of parallel lines, Kant never talked about Euclid’s fifth postulate in his published works at all (though there are comments sprinkled about in his unpublished works). In any case, what Kant says about our a priori intuition is hardly undermined by the discovery that the geometry of space is non-Euclidean (i.e., hyperbolic or elliptic). Andrew Janiak observes:

“Kant highlights the accepted fact that we represent space as an infinite Euclidean magnitude—this can be widely accepted, despite the dispute concerning space’s ontology. […] We do not have a sensation of an infinite Euclidean magnitude.”[5]

Wes Alwan also writes:

“If we discover that the universe is actually, objectively (in the Kantian sense) non-Euclidean when our spatial intuition suggests it is Euclidean, then there is a conflict here between the faculties of understanding and intuition. If you’ve studied non-Euclidean geometry you’ll readily see what this means: the denial of the parallel postulate violates our intuition (unless we model the new geometry within Euclidean geometry but as occurring on a hyperbolic surface); but it does not produce any logical inconsistency. And in fact this is the whole point of Kant calling our perception of Euclidean space “intuition”: I have no other basis for the parallel postulate — I cannot argue for it as following from a principle of logic or arithmetic; nor can I argue about it from some a posteriori discovery in physics about the nature of the world.”[6]

Mlodinow quizzaciously continues:

“Kant, noting that geometers of the day appealed to common sense and graphical figures in their “proofs,” believed that the pretense of rigor ought to be dispensed with, and intuition embraced. Gauss held the opposite view-that rigor was necessary, and most mathematicians were incompetent.”[7]

Nevermind that Kurt Gödel essentially reiterated the very same point as Kant’s, and offered (what he thought was) a demonstration of it in what we now know as the incompleteness theorem (though, to be entirely fair, Mlodinow acknowledges this later on in the book); the point, for Mlodinow, is for us to recognize in the confrontation between Kant and Gauss a microcosmic confrontation between philosophy and science. A conflict from which science emerged victorious over philosophy, physics over common sense, and observation over intuition. The empiricist’s wet dream could not have been better narrated.

Reminiscent of Friedrich Nietzsche’s famous statement that “God is dead!”[8] Mlodinow and Hawking write, in the opening passage of their book The Grand Design, that “Philosophy is dead.”[9] Somewhat ironically,[10] they mean the very same thing, which is that metaphysics is dead. Metaphysics has been confused for a great many things which it is not, so it is worth calling attention to its definition; metaphysics is nothing other than the study of the extra-mental, extra-linguistic, model-independent, objective nature and structure of reality. What the metaphysician wants to know is what the fundamental furniture of reality includes. Where the construction worker is happy to use bricks, metaphysicians want to know what bricks are really made of, and where the mathematician is happy to use numbers the metaphysician wants to know what numbers are. From the perspective of the metaphysician, all the physicist does is offer empirically adequate models of space-time phenomena. That’s it. No amount of empirical data can definitively settle the matter of whether those models are literally accurate, nor can any of those models be the complete story because scientific models (if interpreted literally) presuppose countless philosophical assumptions which cannot be scientifically explored.

Accordingly, Hawking and Mlodinow sketch out, in the book, a view which they call model-dependent realism about science. They write:

“According to model-dependent realism, it is pointless to ask whether a model is real, only whether it agrees with observation.”[11]

In fact, they boldly exclaim that one of the central conclusions of their book is that “there is no picture-independent concept of reality.”[12] In one of the most tantalizing passages of their book they give some indication of just how radical their view really is, and it is worth quoting at some length.

“Model-dependent realism can provide a framework to discuss questions such as: If the world was created a finite time ago, what happened before that? An early Christian philosopher, St. Augustine (354–430), said that the answer was not that God was preparing hell for people who ask such questions, but that time was a property of the world that God created and that time did not exist before the creation, which he believed had occurred not that long ago. That is one possible model, which is favored by those who maintain that the account given in Genesis is literally true even though the world contains fossil and other evidence that makes it look much older. (Were they put there to fool us?) One can also have a different model, in which time continues back 13.7 billion years to the big bang. The model that explains the most about our present observations, including the historical and geological evidence, is the best representation we have of the past. The second model can explain the fossil and radioactive records and the fact that we receive light from galaxies millions of light-years from us, and so this model—the big bang theory—is more useful than the first. Still, neither model can be said to be more real than the other.”[13]

Although somewhat cryptic, it is important that we do not gloss over what’s being said in this passage. What Hawking and Mlodinow are saying is that while some people believe that the big bang theory is true, and six-day creationism is false, and other people believe that the big bang theory is merely closer to being true than the story of six-day creationism, none of these people are correct. As a matter of fact, the big bang hypothesis happens to be a more useful model (given certain hypothetical goals) than six-day creationism and this is the only reason we adopt it in preference to the latter. Although they claim that their view circumvents debates between scientific realists and scientific anti-realists, in reality their model-dependent realism is just a thinly-veiled version of scientific anti-realism!

Now, my chief problem with Mlodinow is not his philosophy of science; he can be an anti-realist until the cows come home and it won’t be any skin off my back. My problem with him isn’t that he thinks we should be empiricists instead of rationalists with respect to objects of intuition like Euclid’s fifth postulate. My problem isn’t even that he thinks that science can license the claim that Euclid’s fifth postulate was literally incorrect (though I do find the suggestion annoying). My real problem with Mlodinow is that I see no way for him to put all of these beliefs together coherently. He cannot on the one hand say that science has shown us that Euclid’s geometry is objectively incorrect, and on the other hand say that no scientific model is ever objectively ‘real’ (by which he means model-independently true). The best he can do, I think, is argue that we ought to abandon Euclid’s fifth postulate when operating within models of geometry which better account for the curvature of space than Euclidean geometry. That’s it, end of story. He cannot even say that Euclid’s fifth postulate was wrong, because the parallel postulate is true within a Euclidean model of geometry! For Mlodinow to say anything which is one iota more philosophically committing than that we should abandon Euclid’s fifth postulate for the same reasons we should abandon Euclidean geometry is for him to wander into utter incoherence.

I want to finish by saying a word or two about this now typical attitude of dismissiveness, condescension, derision and contempt for philosophy among professional scientists, exemplified especially by people like Mlodinow. Although I have no doubt that Mlodinow is a great physicist, it is unfortunate that he has added his (incredibly shrill) voice to the cacophonous choir of scientists grossly overestimating their philosophical aptitudes. What makes his comments particularly irksome is not that I and other philosophers find them disagreeable, but that they are logically irreconcilable. That, to a philosopher, is like hearing the sound of forks scrapped across a chalk board. It really is true what they say; the man who thinks he has no need of philosophy is the one who will be in most need of it.[14] Einstein, whose best friend, it is worth remembering, was none other than Kurt Gödel, was absolutely right when he wrote:

“It has often been said, and certainly not without justification, that the man of science is a poor philosopher. Why then should it not be the right thing for the physicist to let the philosopher do the philosophizing?”[15]

There is one possible reprieve for Mlodinow; although he insinuates fairly strongly throughout his book that Euclid, Kant, et alia were literally wrong about the parallel postulate, he could perhaps backpedal and defend himself by insisting that he never committed himself to the statement that the parallel postulate is literally false. If this is the case, then I owe him an apology for today’s blogging exercise.

[1] Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace (New York: Simon and Schuster, 2010), 116.

[2] It is remarkably silly to say, as people often do, that Galileo was burnt at the stake, or that Marco Antonio Dominis was persecuted for his scientific ideas instead of his vitriolic attacks on the papacy, or that Cecco d’Ascoli was burnt alive for saying that there were people on the other side of the planet instead of his attempt to determine the nativity of Christ by reading his horoscope. There is a modernist myth that the man of science was persecuted in the age of the Church, but this sounds like a phantasmagorical persecution complex. It wasn’t only men of science who got into trouble with the church (it was also artists, writers, poets, theologians, and philosophers), and when men of science did get into trouble it was almost never on account of their scientific work (Galileo is the very notable exception; and serves as the exception which proves the rule). Notice that the same is not true for Theologians. It was, in fact, much more dangerous to do Theology than it ever was to do Science. It is approximately as puerile to say that scientists were, in general, afraid of the Church as to say that Gauss was afraid of secular philosophers.

[3] Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace (New York: Simon and Schuster, 2010), 117.

[4] Immanuel Kant, The Critique of Pure Reason,

[5] Andrew Janiak, “Kant’s Views on Space and Time,” in The Stanford Encyclopedia of Philosophy (Winter 2012 Edition), ed. Edward N. Zalta,


[7] Leonard Mlodinow, Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace (New York: Simon and Schuster, 2010), 117.

[8] Friedrich Nietzsche, The Gay Science, section 125, transl. Walter Kaufmann (1974).

[9] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 5.

[10] Ironic because where Nietzsche as a continental philosopher treated systematic thinking with scorn, Hawking and Mlodinow, as scientists, revel in rigor.

[11] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 46.

[12] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 42.

[13] Stephen Hawking and Leonard Mlodinow, The Grand Design, (Random House Digital, 2010): 49-50.

[14] In particular because to think that one doesn’t need a philosophy is already to have a philosophy (though it is a very bad one).

[15] Quote from Howard, Don A., “Einstein’s Philosophy of Science”, The Stanford Encyclopedia of Philosophy (Winter 2015 Edition), ed. Edward N. Zalta; “Physik und Realität.” Journal of The Franklin Institute 221: 313–347. English translation: “Physics and Reality.” Jean Piccard, trans. Journal of the Franklin Institute221: 348–382. Reprinted in Einstein 1954, 290–323. Note that when taking this quote in its full context it isn’t nearly as complimentary of philosophers, but I’m not sure that Einstein was right about the rest of what he wrote on the matter; I’m merely saying that he was right about this comment.

Banach-Tarski paradox, א Infinities, Infinitesimals, and the A-theory

I will offer an analysis of what is going wrong with the Banach-Tarski paradox suggesting that points, construed as infinitesimal surface areas, are nothing more than mathematically useful fictions. I will suggest that infinitesimals raise the same kinds of modally-prohibitive paradoxes in metaphysics as positing actually infinite quantities does (and for the same or similar reasons), and then consider an argument against the A-theory (in most of its forms) which can be purchased from these insights. I will then scout out some philosophical avenues available to the A-theorist.

The Banach-Tarski paradox is a famous mathematical paradox according to which it can be proved that if you divide the surface area of a sphere into little bits, and simply rearrange the bits appropriately, you can reconstruct two spheres each with the same surface area as the original sphere. In layman’s terms, you can prove (something just a shocking as) that 1=2.[1] To explain how it works, it may be worth calling to mind the various paradoxes associated with actual infinities.

Consider what it would be like to count upwards from -7 to infinity and stop only once you’ve arrived. Even if given an infinite amount of time you would never arrive, because no finite additions can sum up to a transfinite quantity. Subtract infinity from infinity, and what do you have? You have zero, but you also have infinity, and you also have 18.9801 (and every other real number); all of these are not just legitimate answers, they are mathematically correct answers. However, clearly 18.9801 is not equal to either zero, infinity, or anything else! Have a (Hilbert) hotel with an infinite number of rooms, all of which are occupied, and you want to check in an infinite number of new guests? No problem, just move every person from the room they are in (n) to the room with a room number equivalent to two times the original room’s room number (2n). Done; you’ve managed to move people around in such a way as to create an infinite number of vacant rooms without asking anyone to leave. Most of us (who are interested in this sort of thing) know the myriad paradoxes which arise from postulating even the possibility of an actual infinity. It seems relatively philosophically secure that there cannot be an א number of things (where א represents the first transfinite number, not to be confused with ∞ which symbolizes infinity taken as a limit rather than a quantity). If there are philosophically sophisticated caveats then so be it, but the point will remain that there are plenty of examples of things for which having an א number of them is clearly (broadly logically) impossible.

Let’s return, for a moment, to Hilbert’s Hotel, because it’s a particularly useful illustration. Suppose that the guest in room 3 checks out, while all the (infinitely many) other rooms remain occupied. The desk clerk decides that they want every room occupied, so they ask each person in room n (where n>3) to move one room over; that is, from room n to room n-1. That will fill up room 3, but the process will also leave no room empty because there is no room number n for which there is not an occupied room n+1. This works equally well for two dimensional shapes, such as circles; remove one ‘point’ from the circumference of a circle and you may have an infinitesimal gap, but simply move every other point along the circumference over (uniformly) by an infinitesimal amount and, voila, the gap is plugged and there will be no new gap. The trick in the case of the Banach-Tarski paradox is to apply the same reasoning to three-dimensional objects. For the best explanation of this paradox I’ve ever seen, (especially for readers who aren’t familiar with it, please make your life better and) check out Vsauce.

Alexander Pruss has noted on his blog that this result “is taken by some to be an argument against the Axiom of Choice.”[2] However, he argues that you can get the same paradoxical result in similar cases (and even in the same case) without the axiom of choice, so that the axiom of choice should be cleared of all suspicions. I agree (though I’m certainly no expert). Richard Feynman is purported to have said, upon being shown the proof, that “it’s fine you can do it with ‘continuous spheres’, since there’s no such thing. The important thing is you can’t do it with oranges, because oranges are made of a finite number of indivisible parts.” I think he is wrong about oranges (being actually comprised of indivisible finite parts, at least if the ‘parts’ are extended in three spatial dimensions), but his sentiment is appreciably insightful nonetheless.

The problem with the paradox, in my submission, is that it divides the surface of the sphere up into points. However, points on a sphere, like points on a line segment, are infinitesimals. This is precisely why (Aristotelians) say that line segments are not composed of points the way walls are composed of bricks, but, instead, points act as the limits between which a line segment is continuously extended. An infinitesimal is a quantity which is infinitely small. It is non-zero, but it is also smaller than any finite quantity. Sure infinitesimals are useful for doing things like infinitesimal calculus, developed by one of my all time favorite philosophers Gottfried Wilhelm Leibniz, but they remain, I believe, nothing more than useful fictions. To borrow a phrase from W.L. Craig;

“They are akin to ideal gases, frictionless planes, points at infinity, and other useful fictions employed in scientific theories.”[3]

If we are to accept the possibility of infinitesimal quantities in reality, then we will quickly run into paradoxes like the Banach-Tarski paradox (which, quite apart from being obnoxious to the rational intellect, seems to violate the law of conservation of matter and energy). Positing infinitesimals is just as paradoxical as positing sets of actually infinitely many discrete things (where ‘things’ is an ontologically loaded term). I am suggesting that infinitesimals are just as paradoxical as actual infinities, and, at bottom, for the same reason(s). In fact, I have this intuition that every argument for thinking that there cannot be any actual infinities (as opposed to potential infinities, where ‘infinity’ merely acts as a limit), admits of a parody for an argument against the existence of infinitesimals. I’m not sure I can rigorously prove it, but I think it’s very plausible.

It seems to me that there’s something conceptually parasitic about infinitesimals relative to infinities. They each conceptually supervene on each other symmetrically. To visualize this symmetry, consider plotting the function ƒ(x)=  1/x which will look like this:


The distance between the curved line and the x-axis (i.e., y=0) as x approaches (positive or negative) infinity is shrinking (or, at least, its absolute value is shrinking), and approaching an infinitely small non-zero measure. When X is infinite, the absolute value of the y-axis coordinate of the curved line (i.e., the distance between the curved line and it’s asymptote, here being the x axis) is infinitesimally small. This example helps to illustrate the point that the concept of an infinitesimal is bound up with the concept of infinity, so that in the absence of one the other would be inconceivable. That at least motivates the suspicion that if one turns out to be metaphysically impossible, so will the other.

What relevance does this have for the philosophy of time? Well, consider that on the A-theory there is such a time as the present. How long does the present last? What, precisely, is its magnitude, its duration? Let’s consider the following argument:

  1. If the A-theory is true, then the present is either infinitesimal in duration, or it is finite in duration.
  2. The present cannot be infinitesimal in duration.
  3. The present cannot be finite in duration.
  4. Therefore, the A-theory is false

Premise 3 can be established with Leibniz’ argument against the (logical) possibility of a physically indivisible element, or ‘atom’ (in the etymologically literal sense). For anything extended in three-dimensional space, however small, it will always be logically possible for me to divine it in two, even if I am physically incapable of doing so (due to some constraint, such as not having the appropriate equipment for the job, or maybe not even being able to develop any tool which could do the job). Physical impossibilities are not (all) logical impossibilities, and logically there is no constraint on how many times I could divide an object extended in space. To say that there is an object extended in space which is not logically possibly divided up into smaller constituent pieces is, according to Leibniz, incoherent. The exact same argument, mutatis mutandis, works against there being chronons (i.e., atomic chunks of time).

The denial of premise 2 is absurd given our observations that positing infinitesimals leads to modally unconscionable paradoxes like Banach-Tarski.

Ways out: I see four ways, not all of them equally viable, for an A-theorist to escape the conclusion of this argument.

First, they could challenge premise 3 on the grounds that, if there are chronons, then by definition they are entities which cannot be physically divided. The suggestion would be that the prima facie absurdity of a Chronon de dicto doesn’t entail the impossibility of a chronon de re. This dangerously dislocates rational intuition from epistemic reliability, but I can imagine extreme empiricists embracing this response.

Second, they could challenge premise 2 by arguing that positing any more than one real infinitesimal of any kind might be problematic, but that there’s no way to derive similar paradoxes from positing a maximum of one infinitesimal. In other words, perhaps paradoxes involving infinitesimals only arise when there are n infinitesimals, where n ∈ ℕ, and n>1. Multiply an infinitesimal by any natural number, or even a transfinite number, and you will still get an infinitesimal result, so it seems harder to show that from one infinitesimal you could derive some kind of contradiction.

Quick thought: Perhaps if there are rules/axioms such as (i) no infinitesimal can be larger or smaller than any other infinitesimal, but (ii) anything (other than 1) to the power of itself is larger than itself, you could derive a contradiction by taking an infinitesimal X, running it through Xx=Y, and then asking whether Y is larger than X, or the same size (it appears to be both). However, I don’t have the kind of facility in mathematics to be able to produce a rigorous proof that even a single infinitesimal would lead to some kind of contradiction or unconscionable paradox. Moreover, it isn’t entirely clear to me what relevance that kind of mathematical paradox would have for the metaphysical consideration at hand. In any case, the second challenge to premise 2 cannot be lightly dismissed.

Third, one could adopt a really wild philosophy of time, such as the Apresentism I wrote about in the last post (thus denying the first premise).

Fourth, one could deny the first premise by adopting what has been called a non-metric view of the present. This is the view preferred by William Lane Craig.[4] I have more than expended my allotted time for blogging and casual writing today, so I will leave this post here for now. I may return to the idea of non-metric present in the (near) future in another post.

[Ha, I don’t presently have time to write more. Get it?]

[1] For fun, check out and try to find the mistake in the following mathematical proof that 1=2 here:


[3] William Lane Craig, “Response to Greg Welty,” in Beyond the Control of God: Six Views on the Problem of God and Abstract Objects, ed. Paul Gould (A&C Black, 2014), 102.

[4] See: Craig, William. “The extent of the present.” International Studies in the Philosophy of Science 14, no. 2 (2000): 165-185.

Two (new?) Versions of the A-theory

I’m not sure if this is truly original, but I have never encountered anyone defending either of the following two versions of the A-theory of time. Possibly because they are so wildly esoteric as to stand on the periphery of intelligibility. To explain them, let’s begin by thinking about the properties of Presentism, and C.D. Broad’s growing-block theory of time. On Presentism the set of things which exist is identical to the set of things which exist right now. Presentists think that there are no real future events (the future is not ‘waiting for us to get there’), and past events have literally gone out of existence. To become present is to become, full stop. On such a view not all the A-properties distinguished by J.M.E. Mctaggart in his famous essay The Unreality of Time are instantiated. It seems that a prerequisite for an event to literally have a property, such as the property of being ‘future,’ is that the event has to literally exist. If a thing literally fails to exist then it cannot literally have properties of any kind. Similarly, on Broad’s growing-block theory of time, things which are present come into existence at the present, and they continue to exist even once they have inherited the property of being ‘past,’ but future events do not literally exist (or have A-properties) at all.

Keeping these in mind, here are two other views. Since I’ve never encountered them in the literature before, I will take the liberty of giving them names. Let’s call them Apresentism and deteriorating-block theory, respectively.

On Apresentism events bear either the A-property of being future, the A-property of being past, or both, but no events bear neither. In particular, no events bear the A-property of being present. This is because the present, on this view, literally does not exist. The category of the ‘present’ is simply a heuristic tool, a useful fiction, but what it picks out is events which have both the A-properties of being past, and the A-properties of being future. All that ever really happens, on this view, is that events go from having exclusively future A-properties, to having both future and past A-properties, and then finally to having exclusively past A-properties. Is this view logically possible? I’m not inclined to think that it is, but showing what, precisely, has to be wrong with it remains a tall order. If one can show what is wrong with it, then I suspect one will be able to show what is wrong with presentism, but this is just a philosophical hunch.[1]

The deteriorating-block theory is exactly what it sounds like. In place of events having the A-property of being present, and then inheriting the A-property of being past, but never literally having had the A-property of being future (so that layers and layers are added on to the world as the present rolls on), this view suggests that events have the A-property of being future, or the A-property of being present, but to become past is to fall out of existence entirely. Thus, the future is real, the present is real, and the present represents a sort of ontological precipice – events which cease to be present cease to be.

[1] Here’s a quick thought – on presentism construed in the most ontologically conservative way, God would, even if Sempiternal, not literally have a past or a history, anymore than any of us would.

Two Too Simple Objections to Open Theism

First, let’s agree to reject dialetheic logics out of hand; it will be taken as a non-starter for me, and, I hope, for you, if any argument were to proceed on the assumption that a proposition can be both true and false at the same time and in the same sense. It may be useful, at times, to proceed as though this were the case (I’m not denying the usefulness of paraconsistent logics), but it certainly cannot be literally correct. Such logical systems do not (and, by implication granting S5, cannot) describe the extra-mental structure of modality.  

Can God know the future on open theism? It is typically assumed that open theism involves a commitment to Presentism about time (according to which future events are not real, and so propositions about the future are not literally true). I am not sure that this is correct, since they may, perhaps, accept the growing-block theory of time instead, but that will land them in precisely the same predicament as Presentism will as far as my following objections are concerned. In any case, the open theist must accept some version of the A-theory other than the moving-spotlight theory of time (or other more esoteric theories of time which will allow for the reality of future events or states of affairs). God, on the open theist view, shouldn’t be able to know the future because there is no future to know.

It seems undeniable that if “P” is true, and if “P⊃Q” is true, then “Q” is true; that’s just good old Modus Ponens. Now, let’s take P to represent the tripartite conjunction: “the state of affairs S1 in the world will entail the subsequent state of affairs S2 just in case God does not intervene in the world at some time between S1 and S2 (inclusive of S1, not inclusive of S2) and God will not intervene in the world at any time between S1 and S2, and S1 describes the current state of affairs.” Let Q represent the proposition “in the future, S2 will be the case.”

Let us say that God knows P, and God knows that P⊃Q. Does God know Q? If not, He has a deficient grasp of logic. If so, then He knows at least some fact(s) about the future.

  1. If open theism is true, God cannot know the future.
  2. Possibly, God can know propositions like “P” and “P⊃Q.”
  3. If God can know propositions like “P&(P⊃Q),” then God can know propositions like Q.
  4. If God can know propositions like Q, then God can know propositions about the future.
  5. If God can know propositions about the future, then God can know the future.
  6. Therefore, open theism is false.

What will the open theist say? The most plausible response open to them, I think, is to deny premise 5. Generally we think of propositions about the future as having truth-makers which are future states of affairs, but it is conceivable that there be true propositions about the future which have, as their truth-makers, nothing beyond present truth-makers. Perhaps P is presently true, while Modus Ponens and P⊃Q are true presently (they may be timeless truths, so we avoid saying that they are ‘presently’ true, even if they are true presently). That might be a sufficient response. A second response might go like this: premise 1 should be restated as 1*: “if open theism is true, God cannot know the whole future,” and premise 5 should be restated as 5*: “If God can know propositions about the future, then God can know at least some of the future.” Obviously 6 does not logically follow from 1*-5*.

Here’s a second argument:

  1. If a proposition is meaningful, then it cannot fail to be true or false (where the ‘or’ is exclusive).
  2. There are meaningful propositions about the future which are not entailed by any presently available truths.
  3. Therefore, there are true propositions about the future which are not entailed by any presently available truths (they cannot all be false, for if P is false, then “P is false” is true).
  4. God is omniscient.
  5. A being is not omniscient if there are truths (i.e., meaningful true propositions) it fails to know.
  6. If open theism is true, there are meaningful true propositions about the future which God fails to know.
  7. Therefore, open theism is false.

The best responses to this argument which I have heard include (i) denying premise 2 altogether, or (ii) denying premise 1. The denial of premise 1 (given our assumed rejection of dialetheic logics) amounts to a rejection of the law of excluded middle (LEM), and that, my friends, is as good as a reductio against open theism. Rather, it is a reductio of open theism! Alternatively, to deny premise 2 (by denying the meaningfulness of propositions about future states of affairs not entailed by presently available truths), seems implausible given the fact that we all apprehend the meaning of sentences like “tomorrow Julie will eat worms in the playground again.” So, we have at least one relatively good, though simple, argument against open theism.

… Maybe there’s time for a quick third: suppose that epistemic justification means something like ‘true justified belief’ (and let’s, for the moment, ignore Gettier cases, just for simplicity). Now it looks like I can know propositions like P:”tomorrow I will finally propose to her,” even though it looks like God cannot know P! That’s another reductio ad absurdam to add to our growing list of reasons to reject open theism.

My mistake; obviously this last argument presupposes the ‘truth’ of propositions like P, but that’s the very object of contention, so my argument runs in, as they say, a circle of embarrassingly short diameter.

As to whether either of the former arguments will work, it seems to me that if the open theism is too deeply entrenched then the open theist will simply bite the bullet and accept the consequences of my arguments while maintaining open theism. However, at least the arguments can act as a warning to others to avoid the philosophical pit that is open theism.