I think Aristotle criticized the number symbolism of Pythagoreans. Is this true?

In the Metaphysics, he stated that Pythogoreans thought that finitude [= odd numbers?] and infinity [= even numbers?] were not attributes of certain other things…, but that infinity itself and unity itself were the substance of the things of which they are predicated. This is why number was the substance of all things.

Does he then go on to criticize this thinking later? If yes, then on what basis he criticized the Philosophy?

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    Aristotle gives a critical review of his predecessors' views, including the "so called Pythagoreans", in book I of Metaphysics. His criticisms can be found there 990a, and later 1092b. However, modern scholars suggest that Aristotle's account of "Pythagoreans" is inconsistent, and some even that he made up the doctrine he "criticizes", see Cornelli.
    – Conifold
    Dec 11, 2021 at 10:10
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    To add to what Conifold said, Leonid Zhmud in his book Pythagoras and the Early Pythagoreans also argues that the Pythagoreans did not really have an "all is number" doctrine, and that "For Aristotle, as the founder of the history of philosophy, the search for precursors, his own and Plato's, was particularly important; not infrequently it led him to very strange historiographical constructions" (p. 432), with the Pythagoreans cast in the role as precursors to Plato's "unwritten doctrine" where numbers played a main role.
    – Hypnosifl
    Dec 11, 2021 at 17:20

2 Answers 2


Aristotle criticizes the Pythagorean (and Cartesian) notion that numbers comprise the substances of material beings* (Metaphysics 1017b16-22):

substance means those parts which, being present in such things, limit them and designate them as individuals and as a result of whose destruction the whole is destroyed; for example, body is destroyed when surface is, as some say, and surface when line is. And in general it seems to some that number is of this nature; for [according to Pythagoreans and Platonists] if it is destroyed, nothing will exist, and it limits all things.

cf. Thomas Aquinas's commentary on this, Sententia Metaphysicæ lib. 5 l. 10 n. 3 [900-1]; "Everything is not Mathematics.".

*Descartes calls material/sensible substances res extensa ("extended things")—things with length, breadth, and width. However, two substances with identical dimensions can be distinct from each other. As St. Thomas commentates, "that which is found to be common to all things (e.g., quantity) and is something without which they cannot exist does not necessarily constitute their substance".

When discussing in Metaphysics M. (XIII) 6. the "Various ways in which numbers may be conceived as the substance of things" (1080b), Aristotle criticized the principle Pythagorean tenet that everything is comprised of numbers:

And the Pythagoreans, also, believe in one kind of number—the mathematical; only they say it is not separate but sensible substances are formed out of it. For they construct the whole universe out of numbers—only not numbers consisting of abstract units; they suppose the units to have spatial magnitude. But how the first 1 was constructed so as to have magnitude, they seem unable to say.

  • "two substances with identical dimensions can be distinct" Do you mean not just internal dimensions but also identical quantitative relations with other things, i.e. distinct things occupying the exact same region of space at the same time? As a side note, some atomists like Democritus similarly believed the atoms had only quantitative properties like extension/position, and Aristotle in part 4 of book III of De Caelo described this by saying they believe all things are "composed of numbers", much like Pythagoreans (mentioned in part 1)
    – Hypnosifl
    Dec 14, 2021 at 23:46

Probably the best indication that we have of Pythagoras's philosophy is the philosophy of Plato. He identified the monad of the Pythagoras with the form of the Good. Also, according to Hippolytus, the Pythagorean's said the monad begat the dyad which begat the points, which begat the limes, which begat the numbers, which begat finiteness and so on. Plato's take on this is described in his Timeaus, which although not taking directly this route, does use the Pythagorean notion of geometry being the underlying physical-mathematical structure of the world. From this, we can see that the Pythagorean use of numbers to describe their philosophy, wasn't purely number-theoretic, but a form of metaphor. We see an echo of this in descriptions of certain religions as monotheisms (mono - one) and of others, like Zorastrianism, as dualistic (dual - two). Merely because a number has been used in their description does not mean that they are focused on numbers per se.

In fact, Aristotle in book 1 of his Metaphysics whilst discussing the Pythagoreans at 990a, wrote:

how are we to understand that numbers and modifications of numbers of all being and generation both in the beginning and now, and at the same time there is no other number than the number that the universe is composed of?

Aristotle, in his critique of Pythagorean philosophy, instead described it as the first mover. This was part of his more scientific approach to Pythagorean philosophy. In this way he was illuminating the aporia in Pythagorean philosophy, as he saw it.

Simply because Aristotle critiqued Pythagorean philosophy does not mean that he rejected it. Critique is neccessary to improve one's understanding as well as the understanding of others. After, Einstein critiqued Newton, which does not mean he rejected Newtonian physics - he was setting out to improve it. Likewise with Plato and Aristotle with regards to Pythagorean philosophy.


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