I once asked a question on a rigorous and formal definition of conceivable, and was told that there is no such thing as rigorous and formal in philosophy or science. Is this really true? I thought philosophy, like mathematics, would be rigorous, but maybe I am mistaken. Also, I would like to read some texts where philosophers talk about this topic.
Philosophy is usually done informally, simply because the concepts involved do not tend to yield to simple formalizations. Philosophical terms are often derived from natural language terms, which by nature are ambiguous and vary in their intended meaning between different people.
However, this leads to no end of disputes. Philosophers often cannot agree on the precise definitions of terms, nor can they easily determine whether they agree on definitions or not (i.e. whether they are using the terms in exactly the same way). It's not possible to determine precisely what the points of disagreement on definitions may be, without a lot of discussion, and the outcome is usually still debatable at the end. Mathematicians do not have this problem, because they do rely on rigorous formalizations.
So, in philosophy as in mathematics, it is desirable to have formal, rigorous definitions that all can agree on, or at least that all can interpret unambiguously. It's simply not, usually, feasible.
"Conceivable" is a term that we may interpret as, "able to be conceived by a (human) mind." It is not normally formalized. But it's not impossible that it could be, in the future. To do so, we would have to begin with a formal model of what a mind is; the brain is a physical system subject to formal, mathematical physical laws, so creating a formal model of the mind may be possible. Then, working from this formal mind model, we would have to come up with a formal definition of what the mind is conceiving of at a given time, that is not too far off from the natural language word, "conceive" - close enough that we could get away with calling the formal definition by the same word. From this, "conceivable" could be understood as what the mind could conceive of in any of its legal mental states - the union of what it does conceive of in each particular mental state.
All this would be an ambitious research program, but the benefit would be having clear, rigorous terms for mental events, even if they depart slightly from previous usage. We would be able to rigorously say which things are or are not conceivable within our mind-model.
Rigour: strictly according determined constraints.
- In science, rigour is possible, being the scientific method the constraints according which the process can be considered to produce scientific knowledge. Notice that scientific truth scope is empirical (that is, physical).
- in philosophy, there's no equivalent. Notice that philosophical truth is not only about empirical facts (physical), but also about non-empirical facts (metaphysical). Kant's goal with his Critique of Pure Reason was to propose the agenda to develop a method of making metaphysical knowledge following to a precise method, perhaps the scientific method. But his proposal has not found continuity in contemporary philosophy, AFAIK.
Formal: having the quality of a formal system. A formal system is essentially the description of some discipline in terms of axioms and concepts, which allow a specific calculus that allow further logical inferences. A formal system usually depends on a formal language, which avoids ambiguity.
- Most disciplines of science have the structure of formal systems. Thermodynamics has laws. Chemistry has concepts and laws that depend on them. Quantum mechanics needed to develop new forms of mathematical concepts. Etc. Naturally, all are expressed in formal languages.
- Excluding particular exceptions, branches of philosophy are moreover informal. There are not many accepted formal philosophical systems, not because nobody attempted to propose them, but because it is complex to agree on them, given that philosophy depends on multiple subjective elements.
The case of Logic and Mathematics is special. They are usually considered formal sciences, but it is also said that they are part of metaphysics, which is part of philosophy. In any case, both are considered formal systems, where rigour is necessary.
Regarding "conceivable", it depends on the philosophical context where it is used. In general, it is not a relevant philosophical term, as causal, necessary or contingent. For it to be formal, it would need to subscribe to some formal system. However, a term, by itself, cannot be rigorous.
Rigorous and formal are really mathematical terms. Rigour means holding to a strict standard implying no inconsistencies or contradictions, by close attention to criteria for logical consistency, as well as to all relevant evidence and possible differences of interpretation.
Formal in the sense you use it, clearly refers to as in a formal proof. A finite sequence of axioms, or inferences from previous axioms, that together show what is proved. It differs from a natural language argument in that it is rigorous, unambiguous and mechanically (eg computationally) verifiable.
Is there a set of relevant criteria to judge conceivable by? Is there a set of agreed assumptions, or formal systems, from which to consider the term?
'Conceivable', capable of being thought or supposed. We come to know how to use it by simple examples: imagine this -; suppose a situation where -. The language game can begin simply, consequences, counterfactuals, etc.
Can humans conceive of what life would be like for a being consciously living in five dimensions, with some power of movement in all of them? We can use clues, hypershapes, 3D 'slices' that grow and change as higher dimensional shapes move through them. And we can imagine being in 'Flatland', where we could look and see their whole digestive process at once, and move from point to point in ways that seem impossible to the 2D beings. We can even picture the perspective of a being looking across many parallel lives at once. But are we really conceiving of living in higher dimensions? Can we? Can a neuron, really conceive of the brain it is part of..?
So we begin the 'imagine this' game with simple things. And we can go to things we can kind of get a handle on, that we think we can imagine understanding. But it depends a lot on the mental tools to work with, and on the mind doing the conceiving. There seems to be a process of 'getting a handle' on what is being conceived of, like having a 'full picture', being able to get insights and make deductions. It's not easy to say when a difficult thing has been, or when or by what standard it would become so. All this is a marker that it's not a good term to relate to a 'rigorous and formal' discussion. It's too elastic and flexible.
You might like this discussion of how the question 'Why?' gets it's power: "Why ask why" and its scions I would say 'conceive' has a similar power of strange-looping, and self-reference, that leads to tangled hierarchies, or 'inconsistencies'.
Also this, on how the game is begun: According to the major theories of concepts, where do meanings come from?