First of all, I’m a mathematician, not a philosopher, so I apologize in advance for any oversights or if my question maybe isn’t too relevant in philosophy.

When reading popular philosophical content, I often encounter arguments that presume a certain order whose existence isn’t clear to me. For example, Sam Harris argues for objective morality by saying things like, "consider the worst possible state, where everyone is always in the worst possible pain, such that any other state of the universe would be better." The argument then continues based on this premise. I've seen similar arguments in various contexts, such as value hierarchies, where people claim there must be something at the top of the hierarchy, or in discussions about free will, where actions are explained with other actions.

My issue is that people often seem to postulate that such an ordering (e.g., when deciding which state of the universe contains more suffering) is transitive or that it has a smallest/largest element.

This is not at all clear. For example, in rock-paper-scissors, there is a well-defined ordering between the three actions: f.e. rock beats scissors, scissors beat paper. However, this ordering isn’t transitive because rock does not beat paper. So, even though we can compare the options pairwise, there isn’t a worst or best option. And even if it were transitive, there still wouldn’t need to be a single extremum (e.g., there is no largest number).

So I’m wondering, how can philosophical arguments account for these potential issues with assumed orderings? Or is this just something that needs to be postulated and so there isn’t much more to be said about all this.

  • Its not clear whilch putative transitivity fails for Sam Harris
    – Rushi
    Commented May 29 at 4:10
  • 1
    Transitivity is typically apparent or plausible when such arguments are made, so it goes without saying. As for best/worst, it is often a rhetorical exaggeration. Harris's point can be made by saying instead that states with more misery are objectively worse, but "worst possible" makes it more vivid. Considering "worst case scenarios" and extreme cases is often done for sharpening intuition and greater rhetorical effect in heuristic discussions of all sorts. When the "greatest" is essential, there is usually a 'construction' of sorts, like 'summing all perfections' in the ontological argument.
    – Conifold
    Commented May 29 at 7:13
  • Maybe this is just a different way of saying the same thing you are arguing but it seems to me that there's also a problem of 'dimensionality' (for lack of a better term.) If I have a box that is a cube where each edge is on meter and have another elongated box which has the same volume, any ordering (or no ordering) are all valid options.
    – JimmyJames
    Commented May 29 at 15:27
  • 1
    The relation you focus on seems quite similar to en.wikipedia.org/wiki/Condorcet_paradox, "first discovered by Spanish philosopher and theologian Ramon Llull in the 13th century, during his investigations into church governance, but his work was lost until the 21st century. The mathematician and political philosopher Marquis de Condorcet rediscovered the paradox in the late 18th century"
    – J Kusin
    Commented May 29 at 20:36
  • 1
    I don't know if my question on the Economics forum is relevant to yours: economics.stackexchange.com/questions/40312/…. In my view a utility curve is subjective, not objective, and an economist cannot measure utility. But the economist will argue that the behavior of economic agents shows revealed preferences. I say this requires a subjective inference, concerning the behavior of other agents, drawn from one's own subjective judgment. Utility curves are just made-up models so economists can play math games. Commented May 30 at 17:03

10 Answers 10


Well, you raise a good point that I've also thought about.

That said, there is a typical argument for why preferences would correspond to maximizing an expected value: the Von Neumann–Morgenstern utility theorem. In short, if an agent's preferences for different outcomes satisfy certain axioms - completeness, transitivity, continuity, and independence - then the agent acts as if he is assigning a real number to each possible outcome and seeking the maximum expected value.

Why can't the agent have a rock-paper-scissors preference relation? Well, suppose that agent 1 has rock-paper-scissors preferences, and currently has a rock, and agent 2 offers agent 1 paper in exchange for agent 1's rock plus a penny. Then agent 2 offers agent 1 scissors in exchange for agent 1's paper plus a penny. Then agent 2 offers agent 1 a rock in exchange for agent 1's scissors plus a penny. Every time agent 2 goes through this cycle, agent 2 makes a profit of 3 pennies and loses nothing. So repeated indefinitely, agent 1 would lose an unlimited amount of money. This shows the weakness of agent 1's preferences.

I think the biggest problem with the Von Neumann-Morgenstern axioms is axiom 1 (completeness), which says that given any two alternatives A, B, the agent will always prefer one over the other or be indifferent between them. It seems possible that the agent could instead say, "I don't have enough information to tell" - which is neither indifference nor preference. This would result in a partial order over alternatives rather than a total order.

The difference between "I don't know if A or B is better," and "I am indifferent between A and B," is that in the latter, if you have A and someone gives you a penny, you'd be willing to trade A for B. But in the former, you are unwilling to trade, because you can't be sure you aren't being cheated.

  • 3
    "That said, there is a typical argument for why preferences would correspond to a total order: the Von Neumann–Morgenstern utility theorem." That's precisely not the case, since axiom 1 postulates the totality of the (pre)order...
    – Plop
    Commented May 29 at 12:20
  • 5
    Isn't there considerable evidence that individual's utility isn't transitive (i.e. people behave "inconsistently")? Do the other axioms hold of real people? But even if we are happy with individuals having a well-ordered utility function, why is there an ordering on all individuals? Surely any comparison of the real numbers assigned by the utility theorem is arbitrary? Commented May 30 at 7:34
  • I seem to recall coming across studies indicating real intransitive preference in individuals, if I can find them I will post them as a comment, as I think they are worthwhile to consider.
    – tox123
    Commented May 31 at 1:07

Philosophers are not scientists or mathematicians, except incidentally, and what we do as philosophers is not science--it's often pseudo-science. This is not a bad thing. Philosophers frequently create new sciences, or innovate beyond the boundaries of current sciences. But philosophical statements don't have the same level of rigor as mathematical or scientific statements. There typically isn't the pre-existing language or structure to allow for it.

With that said, when a philosopher uses mathematical language or concepts, it's perfectly legitimate to challenge them on places they fall short--there's no magical philosopher's exemption. In the example case, Mr. Harris arguably needs to address the rock-paper-scissors objection directly, or to explain why it isn't an issue. Addressing objections is a key part of how a philosophy can become a science.

It's true that many philosophers use math primarily as a metaphor, and/or deliberately advance arguments that they know are incomplete or inadequate--Plato arguably falls in this category. But the purpose in that case is to encourage you to object. Philosophical arguments are meant to be tested.

  • 1
    This is an evasion.
    – Corbin
    Commented May 30 at 16:18
  • @Corbin in what way? Commented May 30 at 16:25
  • Well, are orders in philosophy somehow exempt from mathematical reasoning? This answer says that challenges exist but doesn't address the presented challenge.
    – Corbin
    Commented May 30 at 16:46
  • 1
    I thought it was implied, but I've reworded to make it explicit. Commented May 30 at 23:28

It is common that many "philosophical" words have been said exceedingly naively. This is a fact which characterizes also the concept of order. Though we have examples of strict philosophical word about order as the one below.

At Proposition 7 in Proclus' Elements of Theology (Στοιχείωσις θεολογική) we read

Everything that is productive of something else is superior to the nature of its product.

This Proposition by itself imposes an order relation on the set of beings based on the criterion of productivity. Of course this criterion is not an exhausting one, for there are non producted (~non created; without cause) beings (this is said specifically in Proposition 11), so the realtion of productivity is not an order relation. However the relation of superiority to the nature is a completely proper relation for our task and this is something said by Proclus:

For either it is superior or it will be either inferior or equal.

The sentence shows that Proclus applies the common order relation of real numbers to the set of beings (B) and he calls it superiority to the nature (s). Consequently we have the well defined structure (B,s) consciously constructed by Proclus not only because of the above but also because he supports it further in his work.

  • Your exception is valid but does not refute the more general argument.
    – Corbin
    Commented May 30 at 16:19
  • @Corbin Did you read my first two sentences? How do you exactly believe that my answer claims refutation of something general? Secondly I think you have not even read the question. If you try again, you will see that I give an example of "assumed ordering" (as the question states: "how can philosophical arguments...") maintenance and even more by absolutely mathematical-in-philososphy means (the questioner says that he is a mathematician); the concept of structure has a purely algebraic and logical formalization and it can be applied to philosophy, philology, sociology, physics.
    – SK_
    Commented May 31 at 8:01

Or is this just something that needs to be postulated and so there isn’t much more to be said about all this.

That's the answer, more or less. If we have a concept of some sort that admits of degrees or amounts of its instances, then we can think through orderly, unorderly, and disorderly possible variants of those instances. Then we say something to the effect of, "We're going to see how this theory works if we start out from the orderly version," and proceed. For example, John Rawls uses the phrase "well-ordered society" a lot in A Theory of Justice, and he is mathematically savvy enough to want us to think of well-orderings in mathematics, but his in-text meaning is not quite that. But so he says (§23, "Formal Constraints of the Concept of Right"):

... a conception of right must impose an ordering on conflicting claims. This requirement springs directly from the role of its principles in adjusting competing ends. There is a difficulty, however, in deciding what counts as an ordering. It is clearly desirable that a conception of justice be complete, that is, able to order all the claims that can arise (or that are likely to in practice). And the ordering should in general be transitive ... But is trial by combat a form of adjudication? After all, physical conflict and a resort to arms result in an ordering; certain claims do win out over others. The main objection to this ordering is not that it may be intransitive [emphasis added]. Rather, it is to avoid the appeal to force and cunning that the principles of right and justice are accepted.

He also says somewhere that it is not clear how to weigh one experience of pain vs. a thousand experiences of pleasure (or vice versa), and (as I read him) that the very process by which we calculate utilities is subject to the maximization protocol in a viciously circular way (there is a utility-maximizing way to calculate the maximization of utility, except how...?). And Rawls is not alone in thinking of possible incommensurable values. For example:

Elizabeth Anderson advances a second argument for constitutive incommensurability. Her account is grounded in a pragmatic account of value. Anderson reduces “‘x is good’ roughly to ‘it is rational to value x,’ where to value something is to adopt toward it a favorable attitude susceptible to rational reflection” (1997, 95). She argues that in virtue of these attitudes there may be no good reason to compare the overall values of two goods. Pragmatism holds that if such a comparison serves no practical function, then the comparative value judgment has no truth value, meaning that the goods are incommensurable (1997, 99). Because the favorable attitudes one adopts toward goods help to make them good, Anderson’s account can be seen as an argument for constitutive incommensurability (Chang 2001, 49).

Now what of the supervenience/determination or metaphysical grounding/ontological dependence orders, for other philosophical examples? Apparently, it is sort-of-consensus that grounding is not monotonic:

enter image description here

And to circle back to the start of the reply:

Some claim that grounding is a strict order—it’s transitive, irreflexive, and asymmetric (Correia 2010; Fine 2012a; Raven 2013). ... While the view that grounding is a strict-order is the prevailing view, what reasons might there be to think otherwise? (As Bliss & Priest [2018b] note, there are various possible combinations of formal properties that we may wish to ultimately consider.)


Interesting and a good question without a doubt.

First, let us focus on your use of rock, paper, scissors as potential evidence that there is not an "ultimate" order, so long as I understand that as the main concern, correctly. Furthermore, I will provide evidence in multiple accounts for the high probability that such an order exists.

The game, which is an important acknowledgement, for games themselves are typically fiction based, uses the utility of the objects, the things themselves, to determine placement. For example, in reality, it is true that the utility of scissors allows for all paper to be cut, albeit sometimes difficultly if the scissors are dull or the paper thick. The paper does not destroy the rock, rather, it covers it, in fact it is placed on top. Thus, the reasoning is that the rock is out of sight out of mind, or even placed below something that is above, being lower in value from a standard hierarchical standpoint. The rock on the other hand, may bust the scissors, for it is true, scissors cannot cut through rock, yet, can be destroyed by such things. In a way, paper can too be destroyed by a rock, if we take the rock and carve it into a single piece of paper, we can rip and tear it. Regardless, the rules of the game do not account for the fact that in reality, if the method of loss is via destruction, the rock is capable of destroying both the scissors and the paper.

Perhaps we have just understood a fixed, objective path to victory via rock, paper, scissors, under this acknowledgment. Maybe, just maybe, this is partly why rock is almost always listed first when the game is played, for numerically it is the number one option.

Despite this, we can indeed see that what Sam Harris is claiming is relevant via probability. In fact, I personally take his point to an objective stance, one that even he would certainly hold as extreme. I'm certain that we would agree on specific moral crimes, ethical crimes, yet differ on other stances solely due to the potential for morality to infringe upon personal liberties. Why would I suggest this? Simply, it is due to the fact that there are moral and ethical principles that seemingly root themselves into the essence of our existence. I suggest that essence comes prior to existence, not the other way around. For instance, you say that you are a mathematician thus I will use a numerical example:

-The number system is objectively defined, it is fixed. Sure, we can use different theories or equations to try and say that specific numbers aren't what they are defined as, yet, this does not change the necessary innateness of numbers themselves. For if 1 did not equal 1, truly, it would be fairly useless in regard to being a foundational number for all other numbers. Factually, if we take one, and via the law of addition, add itself, we get the next number. This process occurs indefinitely.

-Since one is a required foundational number for all numbers, we see that numerically there is indeed a reason to know that 1 = 1 and not 2. The variations this can be expressed in, are vast, surely.

Taking this into account, morals can be viewed like numbers themselves. Ethics the same way. This does not imply that they are numbers, rather, the essence, the fixed nature of such things, is similar. If we cannot agree that causing intentional unmerited harm is bad, then how do we decipher between that which is intentional, unmerited and even harmful? The same with numbers, if we cannot decipher that 1 = 1, then how are we to argue against 1 = 2? The simple answer is that without some sort of absolute, universal relativity, we would not be able to do such a thing.

To finish off this answer, let us understand that good philosophy, does not necessarily assert, for such a thing allows for belief. Rather, good philosophy, just like good mathematics, utilizes a fixed foundation that proves to be supportive of order, not disorder. While it can be fun, philosophically and mathematically to engage in fictitious hypotheticals, we must utilize true reason, ordered reason, to be a ground. One of my personal favorite methods is to utilize definitions and their literal meanings, just as you may do with numbers. The formula would be:

word/concept = description (Not just accepted but reasonable and logical, for there are many definitions that retain contradictions within. The key is to sort out such things via striving for ultimate order.)

To take it one step further, facts and truth over belief and opinion. For in mathematics, opinions hold no weight. That is until someone puts forth a theory that ultimately attempts to re-define most if not all things about such a field. Even then, we must use reason as the guide and not succumb to novelty. If we can assert with fact, it would be no assertion, rather pure acknowledgement of that which is known.


how can philosophical arguments account for these potential issues with assumed orderings? Or is this just something that needs to be postulated and so there isn’t much more to be said about all this.

The short answer is that in the broadest strokes, philosophy isn't a procedure in the same way baking a cake is. It's a sloppy process. First, philosophy, a lot like mathematics, starts off with intuitions (SEP). Shaping those intuitions into coherent claims proceeds by argument and argumentation (SEP). And then ensuring those arguments are grounded well is the domain of metaphysics (SEP). Thus, presuming time is well ordered and linear, making the case for it, and then showing why it must follow from other first principles goes through evolutions the same way any theory does. When a philosopher feels there is enough foundation, coherence, and sometimes empirical support, they then defend the thesis.

Philosophers have several reliable sources of knowledge to account for things in offering metaphysical explanations (SEP). One strategy to account or explain is to rely on the literature, in this instance, about time (SEP). One might appeal to McTaggart's arguments. Another strategy is to rely on science as a basis of facts, as is done with a naturalized epistemology. Some philosophers have even relied on God and divine revelation, though that is certainly not a method of contemporary, professional philosophy.

And a lot of this process comes as the result of asking questions. As for the ordering of time, being linear, what would it mean for time to be partially ordered? Does the theory of relativity mean all time is linear? If time were circular, what would be the impact on thermodynamics? What does the reasoning correspond to in terms of the Münchausen's Trilemma? What arguments are persuasive?

  • The problematic ordering in the question isn't on spacetime or events but on goodness.
    – Corbin
    Commented May 30 at 16:34
  • @Corbin True. But that doesn't change that ordering of any sort is a measure imposed on physical experience, and is therefore part of intuitions and one's first principles. We intuitively understand the ordering time or betweeness and distance with the same vigor with ordering right and wrong. Ask a child to order the from least to worst the following: eating vegetables, being locked in a closet, and being beaten with a belt. Placing those events in a sequence comes naturally, even in the domain of morality.
    – J D
    Commented May 30 at 16:39
  • Let x < y when x is a subset of y. This is not a physical measure but it is a partial order. On intuition, note that human intuition of time is incorrect; we intuit that it is well-ordered but it is only partially ordered.
    – Corbin
    Commented May 30 at 16:53
  • @Corbin Okay, I'll take your distinction of a poset over a metric space; clearly I was sloppy with my language. But the objection doesn't deliver your from the fact that ordering as a process is part of the metatheory that describes each and every theory with order in it, and those are psychological intuitions that arise from our embodied cognition. And it's not just math. It's logic and programs too, because each is built on constitutive grammars. It is the mechanism of grammar itself that linearizes tokens and complexes thereof. Proof? Inference is partially ordered...
    – J D
    Commented May 30 at 17:46
  • Category? Structure is partially ordered. Program? Execution is partially ordered. Thus, to the extent we make sense and describe the world with linguistic intensions and extensions, order is to be found. On the question of time being correct and incorrect, you fall prey to a Boolean tyranny. Linguistic models such as philosophical positions are better characterized along a continuum of adequacy, a point raised implicitly in the the rejection of the Myth of the Given. Newton wasn't wrong. He was inadequate for putting a GPS system in orbit. Real definitions are always subject to...
    – J D
    Commented May 30 at 17:49

I'd say you have to treat each example on its own, you won't find a common answer. Philosophy is not like mathematics in that it is not built on top of a reasonably small amount of axioms from which all else follows in a humongous building of deduction.

(N.B. in surprisingly many philosophical questions, also, many of the terms are not well defined. Even (or especially?) seemingly trivial terms like "existence" have many interpretations. For morality, ethics and such, there are wildly different streams which use the same or similar terms very differently. And often, it is assumed that the reader somehow magically knows what all the implicit meanings could be.)

"consider the worst possible state, where everyone is always in the worst possible pain, such that any other state of the universe would be better."

This sentence would be at the beginning of a line of reasoning, and the meat of the argument would come in the later sentences. You would treat it as a kind of axiom for the specific line of thoughts that follows.

Hence, we would axiomatically pose that there is a worst possible pain. It does not matter how this is defined specifically, we assume it does exist. In these kinds of discussion about pain and happiness, it is assumed that there is some kind of metric which can indeed be compared or added up; the "state of the universe" would then be simply the total sum over all entities capable of feeling pain or happiness.

The question of whether such a metric could in actuality be defined, how we would scale the values between, say, humans and earthworms, whether we could do the summing in practice, and all of that is not the point here.

It is really not that different from maths - there we also very often hide all kinds of arbitrarily small or large (or even non-existant) sets behind very simple terms. And arguably, once you know that you are in some specific field of maths, you'd probably also become a bit sloppy with no ill effect (i.e., if you know that you are talking about Abelian Group Theory, you might just skip mentioning some attributes of some group/operator/set all the time when it is clear from the context).


Such orders don't exist in general. Philosophers who use them are mistaken at best.

When philosophers use words like "worst", "better", or "possible", they are playing Wittgenstein-style word games with concrete syntax, not manipulating abstracta or designating formal constructions. For the specific example in the question, Pirsig 1974 demolished the idea that "good" can be an objective judgement, meaning that "worst" and "better" are not only lacking a comparison (and thus an order) but that they are fundamentally opinion-based; Pirsig says, "'good' is [merely] what you like."

This isn't limited to English either. There are plenty of natural-language examples, and also constructed-language examples, like the Lojban community's arguments over {mau} ("more"), where it is possible to use intensifiers and comparatives to wrongly infer that some objectively-defined order exists over some collection.

  • Ordering preferences is a psychologically real phenomenon. Pleasure and pain can sit on a spectrum, right and wrong can sit on a spectrum, and aesthetic value can be rendered from repulsive to beautiful. "they are playing Wittgenstein-style word games with concrete syntax, not manipulating abstracta or designating formal constructions. " All language is a Sprachspiel, period; that's the chief lesson from the sections of the construction worker using his words teleologically...
    – J D
    Commented May 30 at 16:49
  • 1
    And the notion that comparing and ordering the right and wrong of a rape of a baby and accidentally stepping on someone is somehow completely subjective seems to miss the primary arguments put forth in sociobiology. There are indeed relative aspects to morality, but on the whole, morality is a human universal, as our morality can be traced right back into the Great Apes, who share similar appraisals of various acts and intensities of right and wrong.
    – J D
    Commented May 30 at 16:51
  • @JD: I'm slightly disturbed that you can't stay on-topic, as well as by your choice of tangents. Anyway, pleasure and pain don't differentiate clearly, morality ain't absolute, and Pirsig himself examined your hope that value judgements are detached from the people making them. Meanwhile, the natural numbers are well-ordered and also unique up to unique isomorphism. Orders are abstracta and must be manipulated as such.
    – Corbin
    Commented May 30 at 16:57
  • Pleasure and pain so differentiate clearly. Fully supporting @J D and frankly, if I had enough rep to downvote you @Corbin I would. The notion that such orders don't exist in general is a dissociated viewpoint and will never, truly, be upheld so long as we live within a cosmos. Understand, that the cosmos itself is defined as a well ordered whole. The literal ultimate quantification for the reason we can type these words is ordered by definition. Truly, take your opinion elsewhere for it does not benefit philosophy in the slightest. Otherwise, change the definition of cosmos universally. Commented May 30 at 17:09
  • Furthermore, let it be known that your attempt at defining something such as morality as not absolute is laughable, for you've simply argued that we require broader quantifications for the absolute than what is put forth, generally, via good and bad deciphers of things themselves. Instead of being concise, you argue for a morality that is purely subjective, whereas the objective approach requires accountability on an objectively universal scale. If this were not to be done, order itself would be depreciated. Thus, you take the lazy route and suggest it does not exist. Commented May 30 at 17:18

Not an answer, but a slightly amusing example: a saint asks you to pick a real number in the interval (0, 1) and he will administer a dose of pleasure to a stranger equal in amount to your chosen number. On utilitarianism, you cannot avoid acting wrongly.


The problem here is that philosophy is primarily interested in value orderings, something which doesn't translate well to maths or sciences. For instance, we can look at something like the Mohs scale of mineral hardness and see that it is an ordering whether we go from hard to soft or from soft to hard. However, once we decide that we want 'softness' — i.e., once we impose a value structure in which 'soft' is good and 'hard' is bad — there's only one ordering. Or if we were to change the rock, paper, scissors game so that rock is worth 3 points, scissors worth 2 points, and paper worth 1 point (points to be given to your opponent if you lose), you've created a valued ordering in which paper is decidedly better (since you stand to gain the most and risk the least).

Math and science orderings are (generally) value-neutral, and thus can be inverted and entangled. Philosophical orderings are generally not, and cannot be. On cannot play the early teen game "kiss, marry, kill" in a value-neutral mode without coming off as a psychopath.

In philosophical discussion we can have confusions and ambiguities that stem from vagueness in weighting: e.g., would be prefer a romantic partner who ranks higher on kindness or on honesty? Or we could even (for some reason) say we prefer a romantic partner who is mean and dishonest. But once we've set our values and until/unless our values change, we have a value ordering that only trends in one direction. This leads a lot of philosophers to reach for an ideal state of perfect valuation or perfect devaluation (everything we do/don't want wrapped with a bow) and to take that as something we should make a trajectory for or away from. Only Nihilism has ever strived to strip all value judgements from philosophical thought, but few people have ever embraced nihilism as positive philosophy (instead taking it as the negation of other philosophical positions they reject as low-value).

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .