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These are geneally seen to be on opposing sides. Arguing from axiomatic is arguing from first principles which have clear and distinct conceptualisation in the minds eye of the arguer.

Whereas arguing from authority is simply an argument from a higher authority than oneself, or a recognised authority by the at least the arguer, and in the best cases by the arguee too, (and in the worst cases entirely unacceptable to him).

But surely to argue from axiomatics, is to also argue for the authority of that particular axiomatisation, as opposed to some other; or to argue for an axiomatisation, when a non-axiomatisation might be safer, as the issue under dispute, when reflected on, shows no inclination to be harboured by such geometrically orientated thought; or to argue for the sociological support - ie by a majority of the cognoscenti - who accepts that axiomatisation in the pursuit of that strand or thread of thought?

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I don't think it is. When you argue from authority you say something like this (wiki):

A says P about subject matter S.
A should be trusted about subject matter S.
Therefore, P is correct.

Example. MSRI agrees that the sum of n from n=1 to infinity equals -1/12. MSRI should be trusted about mathematics. Therefore the sum of n from n=1 to infinity equals -1/12. This is an argument from authority, because the conclusion rests on the assumption that researchers at MSRI know their stuff.

When you argue from axioms you say something like this:

Axiomatic system A says P.
Therefore, P.

Example. PA says that [insert Peano Axioms here]. Therefore: addition on N is commutative.

As you can see, the step where you express your "trust" in the axioms is missing, because it's not needed. Whatever anyone may think of the axioms, this argument is valid. Well, I left out the details and qualifications. PA doesn't say that addition is commutative in the sense that it's not among its axioms; but the commutativity of addition follows from the axioms of PA in a sense which can be made very precise. You may have all sorts of reasons to like or dislike PA, but the argument from PA to P when P is a logical consequence of PA doesn't depend on your preferences.

I think the same applies to all arguments from axioms: they don't depend on anyone's belief about the truth of the axioms. In fact, some of the greats like Hilbert, Carnap, and Curry, have (at least once in their writings) entertained the thought that axioms should simply be treated as meaningless sequences of symbols, which can be manipulated to yield other meaningless sequences of symbols by basic combinatorial processes. (If interested in these things, check out Alan Weir's SEP Formalism article).

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  • I should say that the infinity argument doesn't have to be fallacious. It is of course possible (and some have done this) to give a mathematical proof of the fact. But in the form stated above, it is fallacious. – Hunan Rostomyan Feb 6 '14 at 4:39
  • This is the standard argument. However I'm not so sure that one can omit 'trust' in axioms so easily. Surely if I go to the trouble of learning the axioms and understanding how they may be manipulated or used I am investing at least in the beginning and before I can vouch for myself that whatever authority be it I am learning this from - a textbook, or a class - that they are 'worth it'. And generally, I believe in a text, because it comes from an authoritative source, or if I am taught it, it is taight by an authoritative teacher? – Mozibur Ullah Feb 6 '14 at 4:55
  • My understanding of formalism - which admittedly is pretty bare - is that by thinking of theories without their semantic content one can think about theories in general - and start asking questions about the structure of formal lanaguages. So, rather than formal languages are without meaning, it is that their particular meaning is not of interest in this new type of question. – Mozibur Ullah Feb 6 '14 at 4:59
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    To be entirely honest with you, and if its any help, what you've been saying is what would have said until I began to queston what it meant to argue from axioms too...:). And doubly thanks for the fractional amount of respect :). – Mozibur Ullah Feb 6 '14 at 6:29
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    Yes, I think that is one place. Another is whether your axiomatics captures everything of relevance/importance. A third one is what we've been discussing in a roundabout fashion - which is how easy is it to check a certain result derived from the axioms. This third bit I think also has some axiomatisation in computability theory - but I'm a bit hazy on that. – Mozibur Ullah Feb 6 '14 at 6:32
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There is a semblance between the two forms, but I think it disintegrates by looking at the frame in which each type occurs.

There are both roughly as you note:

A says P

Therefore P.

Thus, both are such that the given argument lack a sentential proof their conclusion.

But that does not make them identically fallacious. Axiomatic proofs appeal to the rules that compose the system of proof. Appeals to authority appeal to a person or institution outside of the rules that compose the system.

Using an example from sentential logic itself,

Therefore P v ~P

This is axiomatically true in sentential logic, because it is an immediate consequence of three fundamental laws (identity, excluded middle, and non-contradiction).

Conversely,

Because Gary Kasparov said Bob won the chess game, Bob won the chess game.

Gary Kasparov does not decide the rules of chess so him saying so does not make it so even though he is an authority on chess. Of course, we can make it true by having him look at the board and tell us that was checkmate, but that's a little bit different than an arbitrary case.


Of course where it gets dicey in real life is in cases where one person may think someone's opinion is axiomatic as it composes the rules of the system and someone else may not think so. The first example that springs to mind is that it is imaginable that someone takes a moral claim's truth to depend on God (voluntarism / divine command theory), so that it is axiomatically wrong because God has said so.

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On this issue, a good recent book is

Penelope Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (2011).

In mathematics, in general, you can see axioms in two ways :

formalism : they are "assumed" sentences; I want to see what happens deriving theorems from them.

In this way, mathematics is the set of logical truth of the form (=> is the connective "if ... then ...") :

A => T

where A is the conjunction of the axioms of your theory and T is a theorem.

realism of some sort, where the axioms are selected because you think that they are true or evident or useful.

Saying that you are "committed" to them by the "authority" of the mathematical community, it seems to me incorrect : see modern interset in non-standard or "deviant" logics, and see the richness of alternative axiomatizations of set theory.

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  • Does Maddy discuss the large cardinal axioms? – Mozibur Ullah Feb 6 '14 at 10:25
  • I don't remember; but see also (from Maddy) Realism in Mathematics, Oxford University Press, 1990 and Naturalism in Mathematics, Oxford University Press, 1997. – Mauro ALLEGRANZA Feb 6 '14 at 10:31

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