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Are there any necessary a posteriori truths? (A.J. Ayer 'Language,…

- - - - Ayer says that this should not distress us when it comes to empirical truths- we should not expect these to be certain. We can treat all empirical truths as hypotheses, albeit very probably ones, and still have good reason to believe them- so this should not be an issue for this sort of belief.
      - However, there is more difficulty in explaining the apparent necessary nature of the truths of formal logic and mathematics.
        
        Where the empiricist does encounter difficulty is in con-
        nexion with the truths of formal logic and mathematics.
        For whereas a scientific generalization is readily admitted
        to be fallible, the truths of mathematics and logic appear
        to everyone to be necessary and certain. But if empiricism
        is correct no proposition which has a factual content can
        be necessary or certain. Accordingly the empiricist must
        deal with the truths of logic and mathematics in one of
        the two following ways: he must say either that they are
        not necessary truths, in which case he must account for
        the universal conviction that they are; or he must say that
        they have no factual content, and then he must explain
        how a proposition which is empty of all factual content
        can be true and useful and surprising.
        
        Ayer examines Mill's solution- that the truths of logic and mathematics are not in fact necessary, but just inductive generalisation based upon a very large number of instances- but rejects this, saying that there are no example which could render such propositions false. He gives two examples with explanations of how they could not become false- '5+5=10' and 'my friend has stopped writing to me'
        
        Moving on to Kant- Ayer examines Kant's account of the distinction between analytic and synthetic truths. Ayer syas that Kant appears to provide two different grounds for the discticntion- one psychological, saying that something is synthetic if the jusgement is not contained in the concepts as conceived by te thinker- so 5+7=12 would be synthetic on these grounds as the concept '12' is not contained in the mind of th thinker when he thinks of '5' and '7' and '+', and one logical- analytic jusgemnt s are those where the priciate is conetianed within the sibejct, and synthetic justements are those wheher ethe predicate applied is not contained with the subject
        
        Ayer rejects Kant's account as confused, the two criteria being different. Instead, he prpesew that we can reutain a sensible dicntionction between analytici and synthetic by defining these difference as "a proposition is analytic when its validity depends solely o the definitions of the symbols it contains, and synthetic when its validity is determined by the facts of experience
        
        Ayer says that despite the fact that analytic truths do not give us any new informatio about the world, they are not uninformative- they draw attention to the wyas in which we use various symbols- they make explicit knowledge that is already im;picity contained in the symbols themselves- this making explicit is useful, despite not giving us anything new as such
        
        Kant on geometry as an a priori synthetic truth- it tells us about the world, yet it appears we can deduce geometic truths a priori- this is true if geometry is "the study of the properties of physical space" (quotation from Ayer, not Kant)
        
        Ayer responjds that geometry is not the study of the propeties of physical space as such- the invnetoj of non-Eucidea goemtieries since Kant;s day shows that geometry does not need to be/ is not about our physical world, but is just aother branch of mathematics
        
        "We see now that the axioms of geometry are simply definitions, and that the theorems of geometry are simply the logical consequences of these defintions. A geometry is not in itslef about physical space; in itself cannot be siad to be 'about' anything. But we can use a geometry to reason about physical space"
        
        Objection- we use diagrams to understand geometry. Ayer respoods that that is only due to the limits of our intellect, not anything systematic.
        
        That solved- analytic propositions are definitions, and so cannot be false- that is why they are necessary.
      - If analytic propositions are just tautologies, why should they seem surprising?
        
        This is just due, according to Ayer, to the limits of our powers of reasoning. Ayer thinks that a being with an infinitely powerful intellect would take no interest in logic or mathematics, as it would all seem obvious.