But ‘dependence’ has a different meaning in philosophy than it does in physiology. My ability to speak is dependent on having healthy organs and a healthy brain. But this doesn’t mean that the concepts of mathematics are likewise dependent on brain function. That is where such an account becomes reductionist.
I understand it is natural to look at these issues through the perspective of evolutionary biology. I already conceded that this may well displace Socrates’ belief in the ‘pre-existence of the soul’. That we have an evolved capacity to speak and calculate I have no doubt. But does the furniture of reason — the conceptual content of such symbols as “=” — come into existence as a consequence of evolution?
I say not — that we have evolved to the point of being able to recognise rational truths as possessing a form of reality in their own right. I think it was this insight which animates the Greek philosophical tradition. In some ways, you can sense that they had a kind of intoxication at the power of reason to discern causation. We’ve become a lot more cynical since.
They are. And Place at least relies on the idea of topic-neutral analysis to justify that equality. Chalmers denies that neutrality. In your parlace, he denies that equality can be applied to brain and mind in the manner you describe as problematic, and thus denies that the two can be held to be equal.
I’m saying something similar - that to hold that mind and brain share an identity then we must admit of a physical object qualities which materialists tend to deny. Which suggests that it is materialism itself that is incompatible with identity theory, unless you employ sufficient reductionism that identity theory itself loses relevence.
The meaning of the symbol does not change depending on context. The symbol means ‘equal’.
Sure, why not? In every case what is required is to stipulate in what way the things being compared are the same. Apples and oranges are the same if comparing things that are or are not fruit or edible. Two different things might be the same when it comes to weight or dimensions.
The concepts of mathematics are thoughts and that they are thought depends on brain function. If there were no brains there would be no concepts of mathematics.
It seems to be a meaningless question―like Russell’s question as to whether “the present King of France” is true or false. I’d say neither true nor false because the question is inapt.
First, the question about the meaning of the equal sign and what equal might mean when applied to something other than numbers are two different questions. The fist has been addressed. Second, two stones being equal need not involve numbers. Putting them side by side or filling the same space might be sufficient depending on the application.
The Declaration of Independence says that all men are created equal. In some ways this is obviously not true. They may vary widely in terms of capabilities. They are, however, equal under the law and with regard to rights.
Hair is a characteristic of mammals. Cats and dogs are of the same taxonomic rank, the class Mammalia. With regard to rank they are equal.
Yes it is about the meaning of the symbol. The symbol ‘=’ means ‘equals’. Equals and what it means is a subject of debate, both in philosophy and in formal logic, and in higher mathematics. What is true of equals is true of the symbol.
If the symbol ‘=’ means ‘equals’ and ‘equals’ is subject to philosophical analysis, then that analysis covers the symbol too.
One man’s act of thought is necessarily a different thing from another man’s; one man’s act of thought at one time is necessarily a different thing from the same man’s act of thought at another time. Hence, if ‘whiteness’ were the thought as opposed to its object, no two different men could think of it, and no one man could think of it twice. That which many different thoughts of whiteness have in common is their object, and this object is different from all of them. Thus universals are not thoughts, though when known they are the objects of thoughts.
It will always strike me as noteworthy that the ideal nature of such ideal objects must transcend the different contexts in which they occur. In the passage quoted from Russell, we are dealing with two subjectivities or two distinct moments of thought. However, I wonder how two men come to share the same ideal and universal ‘object’ through tradition and transmission. How is it possible that, through language and writing, I can be taught something like any scientific theory and understand it in its ideal and universal sense? I mean, in schools and universities we are taught this sort of thing, and it seems that it does not come from a Platonic world but rather that the subject learns it through the act of transmission. Is ideality learnt through language and writing? Or is it an act of the subject that confers it upon what has been learnt? This is a problem because if ideality is given to us in the act of transmission, we must regard this act as capable of constituting ideality; and if, on the other hand, it is the subject who adds something to what has been learnt—that is, ideality itself—then we cannot say that ideality is something inherent to the object but rather to the constituting subject.
First, I’ve said that ‘Platonic world’ is misleading. It implies an ‘ethereal domain’ in which ‘forms exist’. It might be better to say that Plato and Aristotle noticed and formalised something essential about cognition (although the term ‘cognition’ is of course not part of their lexicon.) It’s also why I made the point about ‘small p platonism’, which is not the same as ‘the dialogues of Plato’.
Remember the empiricist dictum ‘all knowledge comes through experience’. The mind is tabula rasa, a blank slate, on which concepts are inscribed by experience. I think that is deeply embedded in the way these ideas are thought about. This is the subject of an essay I refer to by French neo-Thomist philosopher Jacques Maritain, The Cultural Impact of Empiricism. And it’s largely unconscious, in the sense that it’s so much part of accepted knowledge, that we see it as simply the way things are. We’re accustomed to looking through it, not at it.
To get back to your question, then - you would never say, ‘what do you mean, “7”’’ You might say ‘are you sure it was 7?’ or ‘I don’t think it was 7’ but you will never mistake 7 for meaning anything other than 7. Hence the appeal to measurable and observable fact that is basic to empiricism and to the success of science. But what number is, is still subject to great controversy. And why? Because number is immaterial in the sense of something grasped only by the mind, but real in it is the same for all who can count, thereby undermining the idea that only material particulars are real.
But that already implies that I know what ‘7’ means. It says nothing about how the ideality and universality of the object are “given” to us. How is ideality constituted? Is it given with the object itself? Or is it constituted by the subject? In the first case, we must accept Platonism; whereas in the second, we must accept idealism, according to which consciousness constitutes ideality.
You haven’t addressed the issue. Grasping does not give us an idea of the genesis of ideality and universality, but rather presupposes it. You say ‘it is the same for all who can count’, but then one must first learn to count, and that is not possible without transmission, language and writing. Which implies that there is something in the means of transmission that constitutes the ideality and universality. Or not, if we accept, on the other hand, that universality is simply given to us through the object.
Not criticism, but comments that may lead (or not) to criticism…
The experience of equality depends on a constant holding onto what is not present; in other words, it implies an absence that is present as an absence: if we perceive equality, it is as though it were repeating itself over time; the permanence of what is equal must occur across different moments in time, where the moments preceding the immediate present are absent. However, for a thing to endure, it must be constituted by moments that are absent; it manifests itself in different successive moments, yet distinct ones, where some remain in the past whilst the perceived universal endures. This is what Husserl calls retention. This is the spatial structure of the universal. The universal is spaced out. This spacing is inseparable from temporal deferral; there is no pure presence of the universal in itself, but always a spacing-in-time. It inscribes itself, takes root and opens up a space in the world. Whether we speak of different intentional acts, different people, different contexts, different particulars or different moments, the structure is the same: universality is founded on repetition where something absent is present. This is the inscription of the universal: it opens up space in history and in the world. That is why ideality and universality (following Husserl) are in the world and are constituted within it. In this sense, the universal is temporal and spatial.