Abstraction in Dummett's Frege

The word abstract derives from the Latin verb abstrahere, which means "to pull away," "to draw off," or "to separate". It is formed by combining the prefix ab- (meaning "away" or "from") with trahere (meaning "to pull" or "to draw").The etymology and evolution of the word reveal how its meaning shifted over time:

Literal Meaning (Physical Separation): In its earliest English uses in the 14th century, "abstract" was used literally to describe something that had been physically removed or drawn away from something else.
Philosophical Concept (Separation of Idea from Matter): By the 15th and 16th centuries, the meaning shifted figuratively. To think about something "abstractly" meant to separate an idea or quality from a concrete, physical object.

We can say 'the thing you are thinking about is an 'object' for your thought'. But it isn't the thing itself. It is something abstracted from that thing. 

 What if you are thinking of something abstract? Then what you are thinking about is something abstracted from an abstraction. This may be useful. It may not. A lot depends on who is doing the thinking and for what purpose. 

Dummett in Chapter 7 of his 'Frege- Philosophy of Language' writes. 

Abstract Objects QUESTIONS SUCH AS whether or not there are any abstract objects, what abstract objects there are, what abstract objects are and how we know that they exist, what is the criterion for their existence, where the dividing line comes between concrete and abstract objects-all these are modern questions.

They are stupid questions. Your thoughts about food aren't food. You can't eat them. But by abstracting away from what concrete examples of food may enable you to find new sources of nourishment. I discovered, quite recently, by thinking about human food- viz. idli- that dhokla too may be edible for our species because it has similar ingredients and is cooked in the same way as idli. True, Gujaratis eat it, but maybe, through 'convergent evolution', their digestive system has become similar to our own.  

At first sight, such a contention appears ludicrous: one might well think such questions to be as old as philosophy. But the fact is that the notion of an 'object' itself, that is, the notion as used in philosophical contexts, is a modern notion, one first introduced by Frege. As we have seen, Frege's approach to questions of ontology involves a clean break with the tradition which had prevailed in philosophy up to his time, and which is still exemplified by such works as Strawson's Individuals. According to the ancient tradition, entities are to be categorized as particulars and universals. It is characteristic of particulars that we can only refer to them and predicate other things (universals) of them- say things about them: we cannot predicate them of anything else--we cannot, as it were, say them of anything.

Sure we can. Physics is the Michael Jordan of the STEM subjects. Still, we get that for some particular purpose we may group a bunch of things together while keeping separate the labels which we might want to attach to that bunch of things.  I buy a bunch of  presents to put under the Christmas tree. I prepare a bunch of labels to be attached to each. No label is a present & no present is a label. 

The problem here is that sometimes the present is the label (it is obvious that the tricycle is for in toddler not the granny) or the label is the present (Oh! My estranged brother remembered me before he succumbed to a heart attack!) 

Universals, by contrast, can both be predicated of particulars, and also referred to in the course of predicating other things (higher universals) of them.

Anything can be predicated of anything & anything could be said to do such predication. The plain fact is the intensional fallacy may arise when it comes to particulars (Bruce Wayne is the Batman! Oh! Actually, Alfred the Butler had donned the mask to throw the Joker off the scent) and which may, in turn, turn out not to exist (e.g. the Ether).  Some 'universals' may turn out to be misconceived or incoherent. They exist in a merely arbitrary, ad hoc or ipse dixit fashion. Such is the case with 'Universal'. Abstraction is okay because it just refers to a familiar type of mental operation similar to 'generalisation' which is related to 'induction'.

 For Frege... this approach is fundamentally misconceived. Terms (proper names) and predicates are expressions of such radically different kinds, that is, play such radically different roles in the language, that it is senseless to suppose that the same thing could be alluded to both by some predicate and by some term.

Talkers can assign any role they like to the noises they make. Mummy understand Baby well enough. Indeed, the dog too soon comes to understand what role is played by the various noises we direct at it.  

It is true enough that we can grasp the sort of thing which a predicate stands for-a concept--only by understanding the linguistic role of a predicate: but just for this very reason we can never conceive of an expression as standing for a thing of that kind if the expression was incapable of playing that linguistic role.

There was a notion that Mathematics itself forbade certain sorts of Mathematics. Russell thought what we call 'non-standard analysis' was either nonsense or a strict impossibility. The aim of using thought to say what thought could or could not do was misconceived. True, one might say of a particular project that it is like trying to invent a permanent motion machine. But as technology changes that may not always be the case. Still, for the present we may dismiss an entire approach to a problem saying it is based on ignoring a well established empirical regularity.  

Dummett's work was careful and his mastery of the subject praiseworthy. However, my feeling is that there were developments in Mathematics which by the beginning of the 1970s undermined the analytical project. These had to do with concurrency, computability, complexity and categoricity. True if we could find an 'absolute proof' or if there were a 'natural' way of showing P is not equal to NP, this would not be the case. 

Thus far, any abstraction we encounter is done by humans who- it may be- have only limited psychological heterogeneity. Going forward, we may find AIs 'abstract' in ways different from us. It may be that all abstraction is not just a drawing away from the particular, but the addition of something we are unconscious of. For different purposes, different things are usefully added though they are 'virtual' and appear and disappear leaving no other trace after they have done their job. It may be that the universe itself is like this.