In his third letter of correspondence with Samuel Clarke, Leibniz outlines his position on the nature of space. Contrary to Newton, and Clarke who defends him, Leibniz holds that space is “something purely relative” (14), that space is not a substance or a thing that exists, but rather the relationship between the objects it seems to contain.

In order to support this claim, Leibniz relies on his principle of sufficient reason, which  states “that nothing happens without a sufficient reason why it should be so rather than otherwise” (14). His axiom is essentially that nothing is arbitrary, and for Leibniz, this means that all things are the product of God’s reason. Leibniz’s argument proceeds by reductio ad absurdum:

  1. Suppose that there is absolute space.
  2. Whence, when God placed objects within space, there was no reason for him to place objects in a particular area of space and not in another, or in a different orientation. Hence, the placement of objects in the universe is arbitrary.
  3. But, according to the principle of sufficient reason, nothing is arbitrary.
  4. But this is absurd, so (1) must be false, and there is no absolute space.

If one accepts the principle of sufficient reason, it seems that one is committed to accepting the argument. Leibniz argues that as long as space is an absolute entity, it is impossible to imagine that the placement of objects within it is anything but arbitrary. The contents of the universe could be rotated by infinitely many degrees inconsequentially, and the reason that it takes any particular orientation cannot be explained causally. Notably, this does not entirely depend on Leibniz’s metaphysical claims about God. Although, for Leibniz, it is through God’s reason that things happen, the principle of sufficient reason can also be interpreted as referring to causality, claiming that nothing is uncaused, entirely spontaneous or arbitrary.

Hence, if space is absolute then the orientation of matter within it is arbitrary. This directly contradicts the accepted principle, and so space must not be absolute.

The fact that such rotation in the universe’s orientation would not cause changes to the physical universe is also mentioned by Leibniz, who asserts that the difference between orientations would be indiscernible (14). As Leibniz points out in a later letter, to “suppose two things indiscernible is to suppose the same thing under two names” (22). If all predicates corresponding to one object also correspond to another (that is, two objects share identical properties), then those objects are identical. This principle is one of second-order logic and can be symbolically represented as:

  • xy[∀P(Px ↔ Py) → x = y]

Since there could be nothing said about one orientation that differs from another, where a and b are any two universe orientations,

  • ∀P(Pa ↔ Pb),


  • (a = b).

The identity of the different universe orientations prompts Leibniz to write that “there is no room to inquire after a reason for the preference of one to the other” (15). This can be taken to mean that the very notion of absolute space is therefore meaningless, since we are left with infinitely many configurations and no reason to prefer any one.

It seems to me that there’s a refutation to this last point. If all of the possible orientations of matter in the universe are in fact identical, then there is in fact reason to for the selection of the current orientation of the universe. For if all universe orientations are identical, then there is exactly one universe orientation – that is, the orientation of the universe is the necessary one. The choice of orientation is necessary rather than arbitrary and therefore it is not in conflict with the principle of sufficient reason.

Source: Leibniz, G. W. and Samuel Clarke. Correspondence. Ed. Roger Ariew. Indianapolis: Hackett, 2000.


Epistemology, Metaphysics

In Book I of On the Nature of Things[1], Lucretius provides arguments for the existence of two main things that exist: matter and the void. Further, he argues that there are no other types of things besides these.

First, Lucretius asserts that matter is known to exist by means of sensory perception. We see matter, we interact with it, and thus it must exist. He argues immediately for empiricism by saying that sensory perception must be the “unshakable foundation” (423) from which an epistemology begins. Otherwise, he says, any talk of what is unseen is meaningless, for it is only in contrast with sensory perception that we can reason about anything that is outside of our view. That is, we cannot make any sense of talk about what is hidden from view without having a view in the first place, and thus what we see in our view must be the case. So, matter exists.

Secondly, we observe some important traits about the matter we see. First, it is always situated somewhere. Second, it can be moved from one location to another. Both of these require the existence of space, what he calls the void, for things always move or are located in something. Lucretius refers back to a few lines before this passage[2], in which he has argued that without space existing between material objects, their force of obstruction would constantly be acting upon each other and there would never be movement at all. He also, in these earlier lines, anticipates an objection on the grounds that fish swim through water seemingly unobstructed. Here, Lucretius argues that water must give way and create space for the fish, thus in order for there to be movement the space must be ontologically present, in modern terms.

Next, Lucretius argues that there must be nothing else besides matter and void. His reasoning is simply that anything else that one may describe, in an effort to posit a third constituent of being, can be shown to be either matter or void. This is done by Lucretius by asking whether or not this new entity is tangible. If it is, and is “susceptible of even the lightest and faintest touch” (433), then it must add to the aggregate matter of the universe and thus count as matter itself. If it is not tangible, however, then it is nothing but “that empty space which we call void” (438). This amounts to the claim that everything exists is either tangible, and is therefore matter, for everything that is tangible is matter, or is intangible, and is therefore void, for everything that is intangible is empty space, that is, void.

Lucretius’ argument for the existence of matter is an ontological an epistemological argument, stating that matter must exist because we cannot meaningfully talk about anything besides matter without first admitting matter into our ontology. However, it is somewhat anthropocentric in its approach, since it relies on what human beings, or perhaps any sentient being, can talk about or count as existing, not what fundamentally exists independently. In this sense, Lucretius is merely categorizing entities into two groups, based on whether or not they are tangible. Since everything is either tangible or not tangible, due to the laws of non-contradiction and the excluded middle, everything can rightfully be said to fall into one of these two categories.

One can rightfully be sceptical of Lucretius’ claim that everything that is intangible is empty space. The tradition of philosophy before and after him posit many so-called intangible objects which are thought to be more than the void. Among them are the forms, or universals, which are thought by Plato to have greater reality than material things. Meanwhile, talk of God or gods both before and after Lucretius often involves an immaterial but certainly active being. Lucretius goes on, beyond this brief passage, to deny that anything can act upon matter without being corporeal[3], but more work must be done to overthrow alternative thought on this matter.

Yet, we can still praise Lucretius for a simple but elegant approach to two formidable views. The simple approach to an epistemological foundation predates the rampant empiricism of the modern era, and even reminds one of Quine‘s naturalized epistemology in its reductionist (and dismissive) approach. The view meanwhile that space counts as some entity in itself, and is needed for the situation and locomotion of matter is one common in modern science, if still debated. One can easily read contemporary developments into Lucretius’ ancient words.


[1] T. Lucretius Carus. On the Nature of Things. Trans. Martin Ferguson Smith. Indianapolis: Hackett, 2001. Lines 419-440.

[2] Lines 335-345 and 370-383.

[3] Lines 440-444.


Leibniz: The Best of All Possible Worlds

According to Leibniz, the actual world is the best of all possible worlds. He outlines a simple argument for this conclusion in The Monadology, §§53-55. The argument proceeds as follows:

  1. God has the idea of infinitely many universes.
  2. Only one of these universes can actually exist.
  3. God’s choices are subject to the principle of sufficient reason, that is, God has reason to choose one thing or another.
  4. God is good.
  5. Therefore, the universe that God chose to exist is the best of all possible worlds.

The argument can be broken down into three parts. Premises (1) and (2) say that God has a choice to make, for there are infinitely many possible worlds, and only one can be chosen. Premises (3) and (4) state that God chooses things with reason (this is the principle of sufficient reason, that is that all things that occur occur for sufficient reasons which fully explain them), and is good – hence, when God makes choices, God chooses that which is most good, or most perfect. Finally, the conclusion puts these two ideas, concluding that in this choice, as with all other choices, God chose that which is best, the best of all possible worlds which could have existed.

The argument is valid, for if there is are infinitely-many choices, and a good, reasonable God has made a choice of one, then that universe is the best. For, if God chose another less-good universe, then God would not be good, God would lack the idea of some universe, or God would not be reasonable, and some premises would be false. (Arguably, one could claim that the argument is invalid on the grounds that its premises may yield a false conclusion if the goodness of universes were identical, and that God did not choose to make any world actual. Clearly, since we know of an existing universe, the claim that none exists would be false, but it is not logically false and hence the argument could have weakness here. However, we will grant Leibniz the tacit assumption that “a universe exists” is a premise.)

Leibniz explains that the reason for the choice of  a world is in its “fitness, or in the degree of perfection that these worlds contain” (76). Hence, the suitability for a world for selection is a function of its level of perfection, and God, being perfect, would choose that world which is most perfect.

Premise (1) follows from the notion of God. God is, by definition all-knowing – a view defended in §48 – and, hence, has the knowledge of what any configuration of the universe would be. So, if God exists, premise (1) is true. Leibniz argues that God must exist in §§44-45 with the argument that “he must exist if he is possible. And since nothing can prevent the possibility of what is without limits, without negation and consequently without contradiction” (74) then God exists. The argument can be symbolized by modal logic as (◊G → □G),  ¬□¬G ∴ G, where G means “God exists”, and is valid.

One may hope to attack premise (2) with the claim that there could be multiple worlds. For if God is a transcendent being, beyond this world, it would seem that he could create all of his possible worlds independently from one another. It may be argued that God would never create anything imperfect, as the other worlds are said to be, but imperfections exist within the actual world even if they are in order with universal harmony. It may be the case that the entire collection of worlds, including the most perfect one, is itself perfect. It is not clear why Leibniz would deny this possibility, though it should also be noted that in this case, we may extend “world” to mean the set of all universes which God is again supreme over, and say that this set is the best of all possible worlds, which would have to be the case since it is the only set in existence.

Premises (3) and (4) seem less controversial. God is defined as being the “ultimate reason” (73) in §38, and hence God is the sufficient reason for things existing. God’s goodness is explained by his perfection in §41.

Hence, it follows from these premises that the actual world is the best of all possible worlds.


Source: Leibniz, G. W. Discourse on Metaphysics and Other Essays. Trans. Daniel Garber and Roger Ariew. Indianapolis: Hackett, 1991.

Epistemology, Metaphysics, Philosophy

In his Mind and World, McDowell contrasts three positions in his concerns with spontaneity and intentional states in general. These are what he calls bald naturalism, rampant platonism and naturalized platonism, the last of which he defends. What McDowell calls ‘bald naturalism’ I will argue for, but I will refer to it more favourably, as ‘naturalism proper’.

The proper naturalist position counts the mind as part of the world, as McDowell wants to do. What a naturalist proper cannot coherently talk about is the Kantian subject. Instead, there are only objects. For those who would say that things are only objects in the sense that they are the object of some subject, then we may use the word ‘substance’, in the earlier Greek sense, to talk about physical things. Thus, the naturalist position is that everything is a substance; in Heidegger’s terminology, everything is essentially present-at-hand[1] and though further supposed properties or features of substances (readiness-to-hand, the existence of Dasein, the mind, intentionality, subjectivity) are reducible to that raw substance. In contemporary physics, that substance is identified with particles, though the details may be somewhat more complex than that[2]. The manifest image of the world[3], the normative space in which intentionality and morality are thought to lie, is composed of that same physical substance that makes up the brains of those who experience and project outwards.

So, for the naturalist proper, the space of reasons is not a realm outside of the natural or physical world, as it is in rampant platonism, nor is it an autonomous space within the world as it is in McDowell’s naturalized platonism. Rather, all rationality, normativity and intentionality is identical with brain states. There is no one space of reasons with which human beings have a connection; rather, each human being can be thought to have their own space of reasons. This space is not especially private, though. It can be seen physically by examination of the neurological states of the brain, and with the proper technology, those states may even be translatable into intelligible images. In another sense, it is not fully private, since glimpses into its content can be seen through behaviour.

‘Content’ needs qualification in that last remark. Intentional mental states are mental states that are about something else in the world. But they are neither a metaphysical link through some imaginary (or real) space to other substances which they are about, nor are they really representations of the substances they seem to be about. They are not the former because such space doesn’t exist in any important way. They are not the latter because, of course, the neurological features of our brain do not form a structure that mimics the outside world. Rather, they form a structure that allows us to have a mental state (which is identical with that brain state) such that we think we have a representation of the substance we are thinking about. So, when I close my eyes and picture the CN Tower, I do not obtain or summon some representation of that object. That object may be gone, moved, or different than I remember. Instead, my brain generates a certain arrangement such that it looks like the CN Tower to itself.[4] All of the qualia, ‘what it’s like to be’[5] and private world phenomena that philosophers expect to find in a separate mental world are  merely what the brain’s activity looks like from the point of view of that very brain.

Against the rampant platonist, the naturalist’s response is brief. Whatever attitudes one has towards the empirical observations with which one is presented, one does not survive without indulging in them. With a proper account of naturalism, any reason to favour a platonic account dissolves. Naturalism proper offers an explanation of intentionality, removes the “spookiness” (McDowell 92) of a separate space of reasons that McDowell complains of. We don’t need to add anything ontologically, and we have a hope to answer the how questions about mental states and intentionality through study of the brain. Spontaneity doesn’t need any explanation here, either, because it is not sui generis, if it’s counted as existing at all. We’re better off with the naturalist account.

Against McDowell’s naturalized platonism, the proper naturalist has to say that McDowell is wrong about naturalism. He argues against his ‘bald naturalist’, saying that “knowing one’s way around the space of reasons, the idea of responsiveness to rational relationships, cannot be reconstructed out of materials that are naturalistic” (McDowell 77). But the naturalist proper doesn’t want to reconstruct a space of reasons, really. The naturalist proper holds that all reasoning, all intentionality, all communication—the entire manifest image—takes place between physical substances across physical substances. The naturalist isn’t committed to reconstructing any intentional state beyond this because she doesn’t hold it to exist. What counts as intentionality for a naturalist proper is evidenced through behaviour, and it consists of a brain processing incoming information. The space of reasons, the finer points of language, art, morality, knowledge and so forth can be talked about as abstractions. They are useful in the processing that the brain must do; they result in digestible inputs. But the naturalist proper is ever-mindful that these are not things in themselves, but that they are always identical with and reducible to the states of the brain. The same holds of spontaneity. The naturalist counts human judgments to be determined (in a deterministic or indeterministic sense[6]) by the physical processes that cause them. Thus, spontaneity can be said either to not really exist, or at least to be denied status as sui generis, as is the case with the other ideas McDowell wants to preserve. Where McDowell wants to say that human beings are free to choose their beliefs, to take their experiences and spontaneously form judgments about them, the proper naturalist denies this. For the naturalist proper, sometimes brain states will have the disposition to respond to the stimulus of a red bench and form the belief that there is a red bench. Sometimes, because of other beliefs, brain states will not have such a disposition and will not form such a belief. Whether or not this happens is not an act of spontaneity, but a function of the states of the brain, including the stimulus received and the beliefs already held.

By positing an autonomous space of reasons, and a second nature with which humans can access it, the naturalized platonist gains little in the way of explanation and much in the way of recalcitrant philosophical questions. McDowell thinks that he gets out of the “threat of supernaturalism” (78) by saying that scientific advancements don’t clarify nature as a whole, but only the realm of law. This step does not work; it only redefines “nature” to include what the naturalist counts as supernatural. McDowell renames nature as the realm of law, says that the platonic heaven, the space of reasons, also exist, and says that there’s this new thing, now called nature, previously called existence, which encompasses all of them. The connection between the realms of laws and reasons is as sketchy as ever, and it is only through proper naturalism of intentionality and spontaneity that the confusion is finally dissolved.


[1] See: Heidegger, Martin. Being and Time. Trans. John MacQuarrie and Edward Robinson. New York: Harper & Row, 1962. p. 98; p. 69 in the original German.

[2] String theories and wave-particle duality in physics play a role here, but they are beside the point. Scientific skepticism about what substances are physically is important. Equally important is the fact that extra-physical phenomena haven’t turned up at all. That is, the naturalist account is the best lead we have.

[3] From Sellars.

[4] …because we are always our physical selves and are never apart from our brains.

[5] From Nagel.

[6] Indeterminism as is afforded by quantum mechanics offers a sort of natural spontaneity, but it is not immediately clear how, or even if, this would impact judgments.

Ethics, Metaphysics

Why you wouldn’t miss free will

Determinism is the view that all things in the physical world are determined by previous physical causes, including human action. Two asteroids collide in space because they’re on a path that brings them together. Water falls off a cliff because gravity pulls it down. And you are reading this because some physical process in your brain has led you to browse the Internet and stumble across this article.

Determinism is the result of applying our scientific notion of causality to ourselves, which we intuitively believe to be somehow immune to the tides of causality. But determinism says that we’re a part of the system.

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