QuestionsCan we be certain of anything?All things are numbers (the Pythagoreans)
Can we be certain of anything?Yes, through reason

All things are numbers (the Pythagoreans)

Pythagoras c. 582 BC · c. 490 BCAll is number.
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What it claims

The elements of numbers are the elements of all things, and the whole universe is a number or a proportion.

Why

Musical scales rest on numbers, which resemble what exists more than fire or water do: so everything would be number.

In short

"All things are numbers (the Pythagoreans)" is an argument by Pythagoras (Aristotle, Metaphysics A 5 (985b-986a), about the Pythagoreans) on the question "Can we be certain of anything?" (answer: yes, through reason). Its claim: The elements of numbers are the elements of all things, and the whole universe is a number or a proportion. According to AskPhilosophy's logic check (Z3 solver), the argument is valid only if one grants a premise the author leaves unstated (P4). AskPhilosophy presents 2 objections, 0 of them with a documented reply, and the original text behind each premise, checked word for word.

Aristotle, Metaphysics A 5 (985b-986a), about the Pythagoreans
Valid with an implicit premise

Pythagoras left no certain writing, and Aristotle attributes this thesis to "the Pythagoreans" ("the so-called Pythagoreans"), never to Pythagoras himself. We do not know what belongs to the master. The argument below is the one Aristotle attributes to them, about 150 years after Pythagoras' death; he writes "they fancied", "they assumed": these are reported impressions, not proofs they gave. Filing: none of our questions asks "what is the world made of?". We file it under "Yes, through reason", because the reasoning starts from mathematics (reason) rather than from the senses. It is not a position Pythagoras took on certainty.

P1The properties and ratios of musical scales rest on numbers.
P2In numbers one thinks one sees more resemblances to what is and what becomes than in fire, earth or water.
P3Numbers are, by nature, first among these principles.
P4 · implicit premiseWhat is first and of which everything else seems to be a copy is what everything else is made of.
Therefore the elements of numbers are the elements of all things, and the whole universe is a number or a proportion.

click a premise to reject it

If you accept every premise, logic forces you to accept the conclusion. So the debate is about the premises.
Checked by a logic solver: the premises cannot all be true while the conclusion is false.
Without the implicit premise: Without P4, the computer finds a case where music rests on numbers, where numbers resemble things and come first, and where nevertheless things are not made of numbers: numbers describe the world without being its stuff.
With P4 the conclusion follows. Without it there is a jump: that numbers resemble things (P2) or describe music well (P1) does not force us to say that things are made of numbers. P4 is written nowhere in the passage: it is our implicit premise, added by charity, and it is where the reasoning moves from "resembles" to "is". It is also where Aristotle's criticisms strike.
Analogy. A score describes a song exactly: each note is a ratio of numbers. But the song is not made of the score: it is made of sounds. The score resembles it closely, it comes "first" for anyone who wants to replay it, and yet it is not the song.

Objections

2

Each objection is a conversation: click to see who replies and where the debate stands.

Aristotle (who reports the thesis) 340 BC · attacks the conclusion

The Pythagoreans construct natural bodies, which have weight and lightness, out of numbers that have none. They therefore appear to be speaking of another heaven and other bodies, not of the ones we perceive. A number weighs nothing: how can it make something that weighs?

no reply
No reply possible: objection raised in -340, after the author's death (-490).
The debate goes on (our summary) We have no reply from the Pythagoreans: this passage is written about 150 years after Pythagoras, the writings of Pythagoreans contemporary with Aristotle (such as Archytas) are not in our corpus, and all we have of their doctrine is Aristotle's report. In the same passage, Aristotle writes that on another point they are "immune from criticism": they do not separate numbers from sensible things. A defender could say they mean "number" as a structure (ratios, proportions), not as something light or heavy. But Aristotle describes their "units" as extended: this reading remains debated.
Aristotle (who reports the thesis) 340 BC · attacks premise 4

The Pythagoreans define carelessly: they take the first thing to which a definition applies as the essence of the thing. It is like believing that "double" and "two" are the same, because two is the first number that is double another. Applied here, it is the same move: taking what comes first, or what resembles, for what the thing is.

no reply
No reply possible: objection raised in -340, after the author's death (-490).
The debate goes on (our summary) Aristotle's passage is about definitions, not about the argument from numbers: linking it to P4 is our reading. A defender would say that the success of mathematics in describing music and the heavens (P1) is more reason than a mere surface resemblance. Pythagoras himself cannot reply: he died about 150 years before Aristotle, and we have no reply from the Pythagoreans in our corpus.