A real realist thought: there are no infinities.
Another thought about this thought: if x supposedly has the characteristic of being infinite, then x does not exist.
Supposition behind this: all entities must have finite boundaries.
What is the status of such remarks?
In the light of them, can we interpret "infinite" in ways that do not have existence implications - e.g. "the series 1,2,3,..... is infinite" means that there is no procedure for establishing an end point, and no more than that? But then, do we not need to avoid the temptation to think "this is what 'infinite' really means"? Why would there be such a temptation if we were immune to the attractions of insidious realism and could use "really" innocuously?
Logic and language: how crude the discussion of the relationship here has so often been. Is logic necessarily an extension of ordinary/natural language - so that it cannot have the autonomy required for it to serve as an independent means of clarifying/reforming such language? If it is an extension, what is wrong with the notion of language turning back on itself to clean itself up?
Imagine a professor kicks off a lecture a by putting a long string of unknown symbols on the blackboard, then, for whatever reason, makes no effort to explain their significance. Could sense be made of them without embedding them in some natural language narrative? If not, does this tell us something about the primacy of language? Could logical notation attain a level of autonomy such that it could be displayed (written/spoken) to purposeful effect absent any natural language contribution, and without the prospect of it ever being translated? Could sense be made here just by linking the symbols involved with other symbols? Could the border be crossed over into mathematics? Could someone get by speaking only mathematics?