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I often come to wonder about a specific, quite abstract question. Since I am not used to writing about such thing, it is very difficult for me to explain, but I will try to present my reasoning by slowly increasing the "difficulty", in three steps:

  • Step 1: I study physics and I therefore often use mathematics. We can think as mathematical logic as being what is it is and that it would be exactly the same in any other universe/reality, i.e. that the concepts of logic is bigger that the physical realization (or physical laws) of our universe. But nothing forces us to think that. Nothing forbids the existence of a universe where 1+1=2 is false, or where the concept of 1+1 is not even defined. If those universe are not forbidden, the fact that the logic 1+1=2 exists in our universe gives us an information about our reality. Now imaging a universe where 1+1=2 is false or were is it not even defined is very difficult because we are used to our universe. But now comes the crucial question: are we be able, in principle, to imagine all the alternative logics or is the fact that we live in our universe constraining our imagination?

  • Step 2: My question has nothing to do with the different mathematical logics, it is just an example. What I am wondering is: is our reality constraining what we can think about or, since thinking is completely abstract, it has nothing to do with reality and we can think about anything we want and go beyond our reality?

  • Step 3: My question has also nothing to do with what us, as humans, can think or not think about in terms "processing power", it is about what is accessible in principle. If I had to reformulate the full question, it would be:

Final question: Does the abstractness of a concept allow it to transcend reality (i.e. to go beyond our reality and any physical laws of our physical, practical, down to earth, universe/reality) ? Or does the fact the we live in a specific universe already restricts the concepts we have access to from the start ?

I know absolutely nothing about philosophy. I hope this question is understandable/interesting/relevant for this branch of SE. I don't necessarily ask answers/opinions, references related to this topic would already be amazing.

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    Graham Priest's book Beyond the Limits of Thought is a survey of the history of attempting to think about this sort of problem in the context of fundamental philosophy.
    – Avi C
    Jun 21 at 20:20
  • "Nothing forbids the existence of a universe where 1+1=2 is false, or where the concept of 1+1 is not even defined." Suppose in our world we have an axiomatic system and rules of inference for arithmetic, at least as strong as the Peano axioms, and which mathematicians believe to be sound. Are you saying that there might be a possible world where even if they used the same axiomatic system and rules of inference, they would get different conclusions? If not, are you just saying that someone might redefine the symbols "1", "2", "+" and "=" to mean something different?
    – Hypnosifl
    Jun 21 at 21:48
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    Concepts are not defined "in" a universe (although creatures to conceive them may not be available there), and $1+1=3$ is forbidden by logic and definitions of concepts involved regardless of the universe. You present your final question as an alternative, but there isn't one. Yes, concepts (obviously) "transcend" reality, we routinely ponder physical laws different from ours, and yes, the laws of our universe, or even peculiarities of our biological constitution, may well restrict the concepts we are able to conceive. They just do not restrict them enough to cover only a single reality, ours.
    – Conifold
    Jun 21 at 22:26
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    Above first main example is not really about logic but the usual axiomatic system of Peano Arithmetic which is constructed upon the classic first order logic. Re your "I know absolutely nothing about philosophy", in philosophy it's critical to have a clear definition for words like "reality/truth" at least from your own POV as a prerequisite to have a useful conceptual analysis. For example, per Kant's transcendental idealism/logic and his definition of "reality", concepts could transcend reality, while for Plato/Spinoza any true idea/concept has to reflect true object thus cannot transcend... Jun 22 at 2:38
  • @Hypnosifl, yes what I mean by "a universe where 1+1=2 is false, or not defined" is not a universe where, us humans, decide on different axioms, symbols, or even a different kind of logic (non-axiomatic for example), but more a universe where such a logic CANNOT exist, or a universe such the very concept of what we would call a logic does not even make sens (a universe that constrains the concepts, and the logic in particular). Maybe the example about mathematical logic is a bad one. Math is just the way I came to think about that, but my question has nothing to do do with math per se.
    – xpsf
    Jun 22 at 8:17

4 Answers 4

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I don't think anybody has reached a definitive answer to the question you asked. However, there have been various attempts to grapple with this problem.

One is Hegel's idea of dialectic. I'm not Hegel expert (who is really) but my understanding is that Hegel thought that abstract concepts don't exist eternally but must come into being through a process of historical manifestation.

Another theory that addresses this question which is not from philosophy is assembly theory by Sara Walker and Lee Cronin. Basically they account for life and ideas in terms of the minimal chain of causal events that are necessary to assemble the structure in question. In this theory, concepts can't exist apart from a physical causal history. There was a recent interview with them where they talk about some of these ideas. What made me think of this from your question was when you said it doesn't have to do with human beings. The nice thing about assembly theory is that the complexity of an idea is an intrinsic property given the laws of physics.

On the transcendence side there are many versions of Platonism which understand concepts to be eternal and independent of physical reality in some sense. If you are a Platonist, it is possible to hold that there might be conceptual truths which could never be knowable by any human mind.

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The relevant meaning of "abstract" is: to extract or remove (something). By which I mean the following. An abstract concept is one that has started with some collection of existing things (objects, characteristics, properties, relationships, etc.). And then several characteristics are abstracted such that we obtain a concept that connects those things.

You start with several existing small fury creatures and build the concept "cat." You do this by abstracting the details that differentiate the cats from each other, but retain the characteristics that specify the category of "cat." Retaining the possibility that a cat can have some degree of damage, anomaly, etc., and still be a cat. As, for example, a cat that has lost an eye is still a cat. A cat is a "small furry quadruped" and so on.

This is how you get abstract concepts. You start with specifics and abstract the characteristics that are not relevant to your analysis in a particular context.

It is not abstraction that allows a concept to be disconected from existence. You cannot transcend reality by removing parts of it.

A very technical example from physics.

A composite object is one which can be identified as having components. A fundamental object is one which resists all attempts at showing it to have components. An atom, for example, is made up of electrons and a nucleus. An electron has no parts and zero size.

In physics, the spin of an object is related to the matrix representation of the rotation group that it sits in. That is my glib explanation of a full year of studying group theory in my master's degree. Here is a wiki page with a very introductory explanation.

https://en.wikipedia.org/wiki/Spin_(physics)

For purposes here, spin comes in multiples of 1/2 starting at 0. So 0, 1/2, 1, 3/2, 2, etc.

In our concepts of physics we find several things that exist in reality. There are fundamental spin-half things. An example is the electron. There are fundamental spin-one things. An example is the photon. There are fundamental spin-two things. An example is the graviton. (Thank you LIGO observatory.) There are fundamental spin-zero things. The example is the Higgs boson. (Thank you ATLAS and CMS collaborations.)

Composite objects of any spin are "easy" to construct. Various atomic nuclei for example have been observed in states with spin in the few-tens sort of range.

However, there are extremely strong reasons to believe that a fundamental object of spin higher than 2 cannot exist. (Sorry, this is where things get massively technical. Beyond reporting what various physicists have claimed, I am unable to give support for this claim.) It is possible to write down a large variety of theories as candidates to match reality, including cases where fundamental objects of very large spin are used. We have never observed a fundamental object of spin higher than 2.

So we have here a situation in which we constructed a theory of a characteristic called spin. We matched up this theory to reality for values 0 to 2. We constructed theories for arbitrarily large values of spin. But we have strong reason to believe no fundamental object over spin 2 can exist.

So any theory of objects of spin larger than 2 must transcend reality.

What does "transcend" mean here? It means that the theory does not correspond to reality, but is internally entirely self consistent. It makes sensible predictions that turn out not to agree with experiment.

But it was not abstraction that produced a non-real theory. It was not removing characteristics. It was extending beyond the range of applicability of the theory that did connect to reality. That is, there is a parameter in the theory. And only certain values of that parameter correspond to reality. Other values are outside of reality.

It is direct to then construct concepts that transcend reality. It is simple. You add characteristics that are not observed. Or you put in parameter ranges that are not observed. These extra features can be attempts to extend physics that fail to be valid. (Spins higher than 2, speeds faster than light, etc.) Or they can be deliberate constructs intended not to correspond to reality. (The invisible pink unicorn being just one example.)

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  • "An electron has no parts and zero size." That's an assumption. We just don't know if it's a point without structure. Its an idea how it could be but probably not is.
    – Pathfinder
    Jun 23 at 12:43
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Many professionals in philosophy of math, ethics, and probably logic too are open to pluralism.

To me this allows concepts to have the free range we want them to have, while never positing they go beyond the universe. Rather, new or more abstract concepts either bring about some new structure themselves (linguistic constructivism maybe) or describe distant but still extant structures we may not have direct experience of yet. Pluralism comes in because even seemingly mutual incompatible concepts can both exist somewhere in the world. And if two different mathematicians want to extend set theory (what I would call abstracting into a larger theory) in different, incompatible ways, it seems like they can. Our world has that pluralistic vastness. It’s no different than both Euclidean and non-Euclidean geometries. Each has their domains. Again, I’m using pluralism to say any new concept or extension (abstraction) of it can find a home in the world, possibly simply by being uttered or thought about.

That is a lesson I learned from Graham Priest’s work (and it may be a wrong one I admit). But by simply a logician finding some story interesting, with no idea of how the story relates to a to the world other than it is a work of literature under study of a logician, that can be used to motivate new logics, like dialetheism.

So to answer your question at the bottom of your post, we’ve witnessed continued abstractions fail to transcend the world, at least if one is a pluralist or some kind of constructivist. And it seems hard to not be some kind of pluralist when Euclidean geometry was abstracted into a greater structure which has both it and other types of geometries. Or, nothing transcended the world when that happened. Maybe this all suggests some greater structure in the world which explains how we can wonder about such a wide range of things when our direct empirical experience seems much narrower, without positing our extra-empirical thoughts somehow broach reality.

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We already live in a universe where 1+1=2 isn't true. Mix two equal amounts of sand and due to better or worse ordering the result may take up more or less space than the constituting parts. Or bring two marginally subcritical masses of uranium in close contact with each other and the result will also be something very different from being twice the mass. Or take periodic structures, rings, torus, globes, etc. that both mathematically and in the real world break with simple addition but allow for 1+1 = 0, because if you take one full period you're back where you started.

It's also quite trivial to think of things that are impossible, just think of what is possible and negate that.

However regardless of what we think about it will almost inevitably have a relation to something that we already experienced in the real world. Like we can mix ideas, concepts, perceptions and impressions in ways that don't match and create things that aren't already there or aren't even possible.

But whatever we create in that way isn't really removed from the reality that we live in but just another product of it. I mean you could sent a friend of yours to a place you've never been to experience things that you've never experienced, let them ponder around it and abstract them to the point where they are no longer recognizable and then present them to you. Then they would transcend your experience but if you'd wanted to you could still trace back the origins to something rooted in this reality. So in a sense yes, but also no.

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  • I didn't mean that we cannot imagine things that are impossible or different/not true in our universe. What I am wondering is if we could, in principle, think of anything, or if the concepts/ideas that we have access to are dictated by the universe we live in. One could argue that the fact alone that such a question can be asked actually proves that we could indeed think of everything and that concepts transcend reality. But this needs more thinking in my opinion, it is not quite clear yet if this works.
    – xpsf
    Jun 23 at 15:23
  • I mean you could argue that every though that we have is made up of combinations of pieces of knowledge and experience and all knowledge is either directly or indirectly the result of our perception of our universe. So yes our universe limits what we can think. So the question is: "Are these building blocks of experience sufficient to build anything?" And that's pretty much unanswerable as if there is something they cannot build we wouldn't know and couldn't know. So if we exclude what we don't and can't know then yeah you probably can conceive any possible idea.
    – haxor789
    Jun 24 at 8:51

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