Are there synthetic a priori truths?
One of the biggest problems with the claim that there could be synthetic a priori truths is that it is likely that it will be difficult to prove the existence of such things analytically, and we are not really entitled to do it synthetically, since that would require us to assume what we are trying to prove. This problem is especially major since it would seem that analytically we are going to have fairly major problems with allowing synthetic a priori. This difficulty arises from the fact that if we are to accept the view of those philosophers who would claim that philosophy is about (at least the majority of the time) how language is used and what words mean, we will probably be forced to accept that 'a priori', which means 'without reference to experience' is entirely incompatible with 'synthetic', which means 'extending ideas about things, and moving beyond a mere analysis of words'. The sort of extensions required with synthetic statements seem to be of the kind that utilise reference to experience to allow us to know things about contingent truths; if statements are not logically certain or impossible it seems unlikely that we will be able to discover whether they are true or false without reference to external, and that means experiential evidence will be needed, rather than internal thought.
Ewing seems to hold an equivalent, yet contrasting view - asking whether 'we know a priori the synthetic proposition that we cannot know a priori any synthetic propositions?'. As lovely and stylish a sentence as this produces, it isn't clear that he is justified in asking it - he gives no reasoning for suggesting that the statement that we can't know a priori any synthetic propositions is synthetic itself. Ewing should have been able to predict the response that obviously there is something contained in the very word 'synthetic' which is incompatible with a priori, not that it is a claim made by connecting the meanings of the words to anything external, and to claim that extensions in meaning could be made simply by thought or reasoning is a pretty untenable position.
Whilst it might seem easier still to hold that there could be synthetic a priori propositions but just that we haven't found any, to state this offhand seems very strange indeed, and it is not at all obvious - especially since the simplest way to prove this would be to show analytically that there is no problem with the marriage of the synthetic with the a priori. Yet it is strange that rather than holding this view various philosophers try to claim that they do have examples of synthetic a priori propositions, and proceed to give examples of things which are very much debatable, and which even the most half-hearted of philosophers could argue with and show that these things do ultimately rely only on definition.
A tangential argument, but one that is fairly interesting, is that brought up by Ewing, who claims that if philosophy is only about linguistic usage, then rules should alter from language to language. Such a claim would seem to deny the possibility of a study of linguistics divorced from any particular language, and this clearly is unfair. He claims that since languages need not have words with exactly the same connotations we would be unable to translate concepts into other languages, and seems to view this as ridiculous. Yet obviously any extrapolation from discussion of bachelors is going to meaningless in a language which chooses not to translate the word in such a way that it means 'unmarried man', and it should be obvious that this would not be an acceptable translation. He seems confused in labelling the statement 'A brother is male' a synthetic empirical proposition about the use of language or not a proposition at all, unless he is saying the perfectly obvious statement, in agreement with the 'linguistic philosopher' who would claim that male is inextricably a part of 'brother'.
A particular favourite example, as brought up by Quinton (to name but one philosopher who seems to believe it to be a good example), of a synthetic a priori is that of the impossibility of an object that is 'red and green all over'. It isn't clear which part of this claim is not to do with the definition of words, but it certainly seems possible to take an analytic view of problem. It seems apparent that 'all over' points at the fact that the thing is completely one colour, and thus cannot be another colour even in part; perhaps the argument is with the terms red and green, in which case you could assure the proposer of synthetic a priori that when we say something is red we are saying it's not not red, and also that red is not the same colour as green. If they argued even further, with the law of contradiction, saying that something could be red and not red, the reply need simply be that by saying something is X we mean that it is also not not-X.
Another example that Ewing uses is that if a first thing is below another, and a third above the second then the third is above the first. Whilst this at first seems as though it always holds, it is possible to imagine a series of objects positioned at progressively more Southerly longitudes, but each above the other. Once we had got to a certain place in our strange spiral, each object above the last, we would find that an object was directly below the first, going straight through the Earth. Given this objection even the philosopher who claims that this is a synthetic proposition would probably crack and respond that we only say that one object is above another because in that case we are defining up in one way, and that Australia is below England only because in this other case we are defining up in a different way. The only other escape route would be to claim that in fact the original proposition was wrong, that height is not in fact transitive and that just because a third object is above a second doesn't guarantee it being above an object below the second. Clearly this isn't a particularly good alternative.
Geometry is an example where many philosophers have tried to claim that we can make a priori assumptions which are based on the structure and shape of the outside world. Kant claims that there is nothing implicit in 'straight line' that would suggest that it is the shortest distance between two points, with no sense of quantity or length contained in the idea of straight. This is a reasonable mistake to make, if you come from the perspective of believing that Euclidean geometry is the only form there is and could be, but this has been found simply not to be so. It has been shown that far from describing the world around us, Euclid's axioms are perfectly arbitrary and that it is possible to create any number of other self-consistent geometries. A straight line drawn on a plain that curves, for example, is not going to be the shortest distance between two points, as taking the route out of the plain will produce a smaller distance. It is only by definition that we find that a straight line produces the shortest distance between two points in Euclidean geometry - and, in fact, find that this is one of the most fundamental axioms - that we define flat plains, with all plains at right angles to each other.
Strangely, Kant seems to attempt to blackmail Hume in an attempt to establish synthetic a priori. He claims that if we get rid of causality, and say that it is merely a psychological concept, based in nothing logical, then we can't have mathematics (presumably claiming that it is based in the same sorts of things). This sort of 'concede causality works or you can't have maths' tactic is hardly one that can overcome the problems that Hume found with causality, and even if we were to accept this line of reasoning, which might be classed a 'baby and bathwater' argument (that we shouldn't reject something if it means throwing out another system which we cannot live without) many would claim that rejecting causality in itself was enough of a sacrifice, and we wouldn't want to preserve it purely because mathematics is threatened, rather than just the presumptions on which we base out everyday life, and regardless it would seem that Kant has a mistaken view of how mathematics is formulated, since most would claim it is the purest of analytic systems.
Frege claims that all numbers are generated by simply adding one over and over again, starting either with nought or one depending on how advanced your system is, and that all the properties of numbers can be deduced from this way that they have been generated. This is clearly, however, not the case. It is not possible to divide number up into odd and even numbers, without first having a concept of the number two - we couldn't say 'An even number is generated every time you add one twice to nought or another even number' since this requires an understanding of what 'twice' means. Likewise it seems unlikely that we could formulate multiplication by saying that you add one a certain number of times, a certain other number of times, since then you would just be saying something like 'add one to nought four times, three times', and clearly this already requires an understanding of four and three. If the only operation we are allowed when performing mathematical operations in the addition of one a certain number of times, we are never going to get anywhere since we will never know when to stop.
Frege even seems to understand this problem, or an equivalent one - pointing out that with many things it is the shared properties of various particulars that are important, yet with numbers there is a problem because we can never easily tell when the special characteristics of a particular number will come into play. This seems to suggest that he did have some insight into the problem above - if we are only ever able to use the properties of the number one, how can we ever think about characteristics specific to higher numbers. Surely the concept of even is meaningless before we have two numbers, just as we can't describe how to raise something to a power before we have a concept of multiplying.
Kant's answer, that 'All mathematical judgements, without exception, are synthetic' does not seem to solve the problem. If this were so then we should only be able to perform mathematics by observation, and only by counting fingers, for example, could we formulate a concept of addition, and we would never be able to have a concept of mathematics apart from objects. Yet if this were so it would hardly prove Kant's claim that these are synthetic a priori, since they would seem entirely empirical rather than from first principles in any way.
If, instead of following Frege's advice, we allow other numbers than one, as Frege probably intended, then we can follow his definition, since it will always be possible given a concept of 7 and 5 to use his technique of simply adding one 5 times given 7 to get to 12. Then clearly we will be able to reach the number 12 using these simple definitions. It seems a mistake to claim that we arrive at difficulties at this first hurdle, of adding a single number to another, as clearly it will always be possible to define numbers in terms of a single addition. Thus we should have no problem with the sum 5 + 7 = 12, since we can decide this merely by definition, if we choose to. The problem arises when we have more complicated equations, such as 5 + 7 = 6 + 6 since in this case it will not be possible to formulate it by definition, since there are two different sums involved. The claim, however, that this must mean the equivalence is a priori in no way provides an answer, and we are never going to get to a satisfactory solution if we claim that we 'instinctively' know each and every possible sum. It might be possible to claim that from the various sums that we do instinctively know we can produce rules by which to work out larger numbers. Even in this case, however, whilst the rules might be a priori, they clearly cannot be synthetic, since they are simply working with our definitions.
Overall, therefore, I see no reason to suppose that a priori synthetic propositions should exist. All examples that are produced seem to be flawed, and the claim that such things could exist whilst there are not currently any known examples would require an analytic proof and one does not seem terribly forthcoming. Many philosophers have found examples of what they believe to be synthetic a priori, and whilst it cannot be seen as conclusive the fact that these can be systematically disproved would not seem to add any weight to their case, and a valid example is yet to be produced that cannot be destroyed by an argument against.