...might in fact be the right interpretation of the proposition... — javra

This I believe is the key phrase toward understanding javra's position on this matter. We must consider "2+2=4" to be a sequence of symbols requiring interpretation, to abstract meaning. And, there is always some degree of subjectivity which enters that process of interpretation. So, when two different people produce two different descriptions, (interpretations) of the meaning which they each abstract from the phrase "2+2=4", we can judge one as a better interpretation than the other. And, if we leave the possibility open, that we can always find a better interpretation, then the question of "the right interpretation" remains unanswered.

Goodness is a value and as far as I can see values can exist only in, be held by, minds

Langauge and the rules of interpretation in semantics don't make any sense unless you have something to fix them beyond your mental states. So "goodness" , like "rationality" isn't just a value inside your mind , it's meaning and truth conditions are determined by both your mental states and that which is external to it (other minds, universals, particulars, God etc)

This is why if someone comes up to me & claims murder is good because that's how he imagines it inside his head, then I would have no problem explaining to him why he has made a mistake here. That's not how we understand "goodness". If he still doesn't get it, then we will just consider him to be suffering from some mental impairment, just as we would for someone who claims 1+1=3 , despite repeated efforts to correct this mistake.

Aye, I don’t want to turn this thread into one of epistemology. But to answer …

I’m sorry but I’m one of those stodgy old-fashioned types who believe that 2+2=4 is true in all possible worlds. I can’t see how a world would hold together if it were not. — Wayfarer

And yet we live in a world where some people, some more fervently than others, believe that a certain 1 + 1 + 1 = 1 (often termed the Trinity) and that it is only due to this state of affairs that the proposition of 2 + 2 = 4 can possibly hold true. Put a philosopher’s or a mathematician’s hat on and one might see a blatant logical contradiction here. At any rate, I take this world where the Trinity is taken to be factual (and hence were a certain 1 + 1 + 1 does equate to 1, rather than to 3) to be one such possible world.

How could 2+2=4 be wrong ? Our mathematical knowledge is more certain than any philosophical argument you can bring against it. If a philosophical view requires us to doubt 2+2=4, then I would rather abandon that philosophical view, than allow uncertainty into mathematics. — Sirius

I've already given reasoning for 2 + 2 = 4 not being an infallible proposition. What you're here addressing is not the reasoning but the conclusion and how you'd react to it. But, as to the conclusion, that there is a possibility - irrespective of how minuscule - of X being wrong in no way entails that X is in fact wrong. Furthermore, just because X can be rationally evidenced fallible rather than infallible does not in any way warrant that one then doubts X. What reason is there to doubt X when X exhibits no inconsistencies - this despite X being fallible nevertheless?

This I believe is the key phrase toward understanding javra's position on this matter. — Metaphysician Undercover

Yes, it certainly is pivotal to what my argument for global fallibilism consists of.

But, again, I find no reason to doubt the truth of 2+2=4 in the absence of inconsistencies. And 2+2=4 is certainly consistent.

I disagree with you, but I acknowledge that no logical argument can prove you wrong. It also seems to me that our difference on this point is vanishing small- as small as the possibility that "2+2=4" is false.

I disagree with you, but I acknowledge that no logical argument can prove you wrong. — Relativist

I understand. Although don't we here then embark into areas of faith, rather then those of belief which can be justified.

It also seems to me that our difference on this point is vanishing small- as small as the possibility that "2+2=4" is false. — Relativist

Yes, I can agree, hence why I consider my belief that 2+2=4 to be categorical - despite it yet being, technically when philosophically appraised, fallible rather than infallible.

Why would you create duality between subjective & objective means ? If God does exist, then his being qua being would both be nondelimited prior to manifestation and delimited via manifestation in the mental & physical world (assume both categories are relative to one another). — Sirius

The subjective/objective difference is simply that an objective means is demonstrable - it can be shown to be true to others. If someone believes they've personally experienced a God, that can help justify his belief to himself, but it has no power to persuade anyone else.

The trick is to stop looking for God and understand he has not only always been with you but he is identical to your reality. — Sirius

I hope you understand that this statement can't possibly persuade anyone that a God exists- and that's because it depends on the premise that a God exists.

People seem to think God is like a pseudo object which exists apart from the universe, which is just superstitious & baseless.

IOW, you don't consider the God of your belief to exist apart from the universe. OTOH, I see no reason to think that anything like a "God" exists in any objective sense. I'm fine with you embracing your belief. I'm certain I couldn't possibly convince you you're wrong, even if I wanted to (which I don't). I hope you are sufficiently open-minded enough to understand why I don't share your belief.

If you want to know God, you just need to think differently of him, or to put it more succinctly, you need to stop thinking of him, as he is beyond concepts and experience as well.

I admire your passion. I hope your belief helps you to do good.

Yes, it certainly is pivotal to what my argument for global fallibilism consists of.

But, again, I find no reason to doubt the truth of 2+2=4 in the absence of inconsistencies. And 2+2=4 is certainly consistent. — javra

There is more to truth than consistency, there is also the matter of correspondence with reality. This is why interpretation is pivotal. To judge whether 2+2=4 is consistent we need a statement to determine 'consistent with what'. The "what" here forms the basis of the supposed reality which we will judge correspondence with.

So for example, in basic arithmetic we might interpret each "2" as signifying two objects, the "+" as signifying an operation of addition, the "=" as signifying equivalence, and "4" as signifying four objects. But more sophisticated mathematicians might interpret "2+2" as signifying an object, and "4" as signifying an object, and "=" as signifying "is the same as". The "reality" which "2+2=4" corresponds with (is consistent with), making it true, is determined by the principles (axioms) which the interpretation is based in.

There is more to truth than consistency, there is also the matter of correspondence with reality. — Metaphysician Undercover

Thanks for that, and I by this am not in any way disagreeing with your reply.

In the part I've just quoted: I find consistency and correspondence/conformity to reality to be deeply entwined. This in so far as reality, whatever it might in fact be, can only be devoid of logical contradictions (for emphasis, where an ontological logical contradiction is a state of affairs wherein both X and not-X both ontically occur simultaneously and in the exact same respect). For example, if reality is in part tychistic then truths will conform to this partly tychistic reality in consistent ways - thereby making some variant of indeterminism true and the strictly hard determinism which is currently fashionable among many false.

It's a whopper of a metaphysical claim that realty is devoid of logical contradictions - although I so far find that everyone at least implicitly lives by this conviction. But, in granting this explicitly, then for any belief to be true, in its then needing to conform to reality to so be, the belief will then necessarily be devoid of logical contradictions in its justifications (which, after all, are justifications for the belief being conformant to reality, or else that which is real). So if a) reality is consistent (devoid of logical contradictions) and b) truth is conformity to reality then c) any belief which is inconsistent will not be true.

As to the truth of numbers, their relations, and what they represent:

If physicalism, maths can only represent physical entities and their possible physical relations (otherwise it wouldn't be physicalism). If non-physicalism, then the numbers made use by maths could in certain situations represent incorporeal entities, such as individual souls or psyches. In the here very broad umbrella of the latter, one could then obtain the proposition that "one incorporeal psyche added to another incorporeal psyche added to another incorporeal psyche can via assimilation converge into one possibly grander incorporeal psyche" - thereby holding the potential of producing the 1+1+1=1 proposition, which will contradict the 2+2=4 proposition IFF the numbers of both equations are taken to represent the same corporeal and hence physical objects. Otherwise, within at least some non-physicalist worldviews, the question which is so easily ridiculed from physical vantages can emerge: how many individual incorporeal beings, such as angels, can fit onto the tip of a pin? With the answer being indeterminable due to the very incorporeal nature of individual beings addressed - creating a deep equivocation of sorts.

The basic general point to all this tmk being in general agreement with your post

So as to not overly focus on Chistian beliefs, I should maybe add that the non-physicalist understanding of numbers as I’ve just outlined it pervades popular culture at large: from the notion that (non-physical) being is one (as in the statement, "we are all one," or the dictum of "e pluribus unum") to the notion that in a romantic relationship the two can become one. With all such beliefs being disparate from the stance that 1+1 can only equal 2 in all cases.

But none of this is to deny that in physical reality 1+1 can only equal 2 - and, by extension, that 2+2 thereby equals (and can only equal) 4 when it comes to physical entities.

In the part I've just quoted: I find consistency and correspondence/conformity to reality to be deeply entwined. This in so far as reality, whatever it might in fact be, can only be devoid of logical contradictions (for emphasis, where an ontological logical contradiction is a state of affairs wherein both X and not-X both ontically occur simultaneously and in the exact same respect). For example, if reality is in part tychistic then truths will conform to this partly tychistic reality in consistent ways - thereby making some variant of indeterminism true and the strictly hard determinism which is currently fashionable among many false. — javra

The point I was trying to express, is that reality is what we make it to be, as in the the op, mind created world. So to correspond with reality means to be consistent with the principles we state as being those which describe reality. This puts logic in a sort of awkward place. We might say that reality is such that it is devoid of logical contradictions, but what this really means is that this reality which is devoid of logical contradiction is the product of a desire to maintain the law of non-contradiction. Ontologies have been proposed in which the law of non-contradiction is not necessarily true. This would mean that these principles give us a different reality.

It's a whopper of a metaphysical claim that realty is devoid of logical contradictions - although I so far find that everyone at least implicitly lives by this conviction. But, in granting this explicitly, then for any belief to be true, in its then needing to conform to reality to so be, the belief will then necessarily be devoid of logical contradictions in its justifications (which, after all, are justifications for the belief being conformant to reality, or else that which is real). So if a) reality is consistent (devoid of logical contradictions) and b) truth is conformity to reality then c) any belief which is inconsistent will not be true. — javra

So this is where things get difficult. Let's say that we assume a reality which allows for logical contradictions. Then, logical contradictions may be consistent with reality. Therefore a true belief may be logically inconsistent. But remember, we create reality by naming the principles which describe it. So if we think it's a better reality, we can insist that contradictions be avoided. Aristotle for instance saw a need to allow for violation of the law of excluded middle to create a reality including potential and possibility, but he insisted on maintaining non-contradiction.

If non-physicalism, then the numbers made use by maths could in certain situations represent incorporeal entities, such as individual souls or psyches. In the here very broad umbrella of the latter, one could then obtain the proposition that "one incorporeal psyche added to another incorporeal psyche added to another incorporeal psyche can via assimilation converge into one possibly grander incorporeal psyche" - thereby holding the potential of producing the 1+1+1=1 proposition, which will contradict the 2+2=4 proposition — javra

This is sort of what Platonism does. A numeral represents an object known as a number, so one object plus one object equals one object. However, the objects each have different values, and the value is represented by the numeral, so we do not have 1+1=1.

So as to not overly focus on Chistian beliefs, I should maybe add that the non-physicalist understanding of numbers as I’ve just outlined it pervades popular culture at large: from the notion that (non-physical) being is one (as in the statement, "we are all one," or the dictum of "e pluribus unum") to the notion that in a romantic relationship the two can become one. With all such beliefs being disparate from the stance that 1+1 can only equal 2 in all cases. — javra

Platonism pervades mathematics. We learn in school the difference between a numeral and a number. The numeral "2" signifies an object, which is the number two. Then what is important is the value assigned to the object. So two really does become one, but that one has a distinctly different value from what the other two each have. And in a fundamental way it is consistent with the physical approach in the sense that the values indicated are the same. However, there is a non-physical object which is added in.

The point about shape, with boulders and cracks, has to do with the relative size of mind-independent objects, and these relative sizes will hold good whether or not they are measured. It must be so if boulders treat cracks differently than canyons whether or not a mind is involved. — Leontiskos

I have thought again about your objections since you raised them again recently. I don't believe they actually refute the points made in the original post. As it is a defense of idealism, I'll refer to Schopenhauer and Berkeley.

Schopenhauer would argue that both shape and color belong to the realm of representation (Vorstellung), which is inherently conditioned by the subject. Shape, while less obviously subjective than color, still relies on spatial and causal relations that arise from the mind’s structuring of sensory data. A boulder rolling into a canyon is a phenomenon, an appearance - and, as such, dependent on the forms of perception (space, time, and causality) that the mind imposes on the raw data (which Schopenhauer designates 'will'). When we say the boulder "has dimensions that are such and such," this statement itself relies on a conceptual framework — one that includes notions of measurement, spatial relations, and causality. A boulder, after all, does not possess or conceive of its own dimensions. It is we perceivers who bring to it the ideas of "shape," "size," or "falling into a canyon." As said in the essay, take away all perspective, any awareness of shape, size and position, and what exists? Again, to point to the so-called 'unperceived boulder' is itself a mental construct, relying, as I said, on an implicit perspective.

As for the universe’s existence prior to minds, Schopenhauer would agree that the world exists as Will, but he would deny that the world as we can ever conceive it — as an ordered totality of objects in space and time — could meaningfully exist without a subject. To speak of such a universe is to again to reintroduce the forms of representation. The universe prior to life, in Schopenhauer’s terms, would be an undifferentiated striving will, not the structured cosmos we now perceive.

Berkeley would agree that minds can know real properties but would reject the assumption that these properties exist independently of the perception of them. What you call "realism" — the belief in mind-independent objects — requires positing an unobservable substratum that supports properties like shape. Berkeley would argue that such a substratum is unnecessary and unintelligible; all that we perceive occurs to us as ideas, and these ideas are dependent on perception. Berkeley doesn't deny that objects behave and appear to be material in nature, but emphasises the 'appears to be', and denies that they exist in some sense externally to that.

None of which is to deny the empirical fact that boulders will roll over cracks and into canyons, and even fetch up in places where Samuel Johnson will be able to kick one of them. ;-)