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This thought started off as 'If we are in a simulation, or if there exist a God/omnipotent being that can change the laws of physics (and everything) as we know it - is it possible for a change to defy reality as we know it?"

The most dramatic example I could think of is 1+1=3.

Ex: One shoe, another shoe, bam - 3 shoes.

This concept of a world where 1+1=3 is really hard to conceptualize for me because it changes everything that I understand the world as - which creates a picture for my original thought process:

Are there rules that even God(s) have to follow when formulating a world? Is it possible for us to be completely sure that 1+1=2 and there's no other way around it? Or, am I having a trouble conceptualizing this simply because I've lived my whole life in a world where 1+1=2 and laws of the universe are possibly more fluid than I had originally imagined?

(Note: Although I am indeed curious if 1+1=3 is possible in a theoretical world, please note that this is just an example for a bigger question)

Jett
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  • One position is at least is that you must distinguish types of knowledge. There is science tific knowledge which most of humanity relies on.There is something else referred to as OBJECTIVE KNOWLEDGE . Now in the case of science nothing is absolute. That is every theory must be falsifiable by definition. Objective knowledge is absolute. The truth value will never change by definition. So what is true must remain true forever. No false today & true 50 years later stuff. So in the case of objective knowledge GOD would not be able to change it by definition. Nothing can change the objective kind. – Logikal Mar 16 '20 at 15:52
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    "Could an [omnipotent being](https://www.iep.utm.edu/omnipote/) draw a square circle? Descartes notoriously answered “yes.” However, the Western philosophical and theological traditions have, at least since Aquinas, almost universally given the opposite answer. The view that an omnipotent being could do absolutely anything, even the logically absurd, is known as [voluntarism](https://www.iep.utm.edu/voluntar/)." – Mauro ALLEGRANZA Mar 16 '20 at 16:01
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    Seems like there might be two aspects to this question, one about whether "a world where 1+1=3" is actually a possible world, or even expresses a coherent or conceivable concept (some philosophers do discuss [impossible worlds](https://plato.stanford.edu/entries/impossible-worlds/)), and second about whether philosophical views about God's omnipotence say that God can change the "laws" of mathematics or logic. Might be a good idea to edit the question a bit to focus on which question you're more interested in, or if you're interested in both split it up into two more focused questions. – Hypnosifl Mar 16 '20 at 18:30
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    1+1=3 is not something that can be in the world or even about the world, it is a mathematical abstraction. So the title question does not really make sense. But we can easily make it true "in" our world by swapping the use of symbols "2" and "3". Even without that there can be a world where putting two things together makes a third thing pop up. There can even be a world where this happens often enough for its intelligent inhabitants to incorporate it into their rules of arithmetic. It doesn't even take omnipotence. But it is a postulated *rule*, not something to be sure or unsure about. – Conifold Mar 17 '20 at 04:49
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    @Conifold - Even in that hypothetical world, ppl would likely have notions of arithmetic matching ours for *conceptual* groupings (such groupings may be the intuition behind sets in set theory). For example if you pick some volume of space at a single instant of time and divide it conceptually into two halves, then ask "how many people are in the left half" and "how many people are in the right half" and the answer in each case is "1", then they would presumably agree this implies that the whole volume contains 2 ppl at that instant, even if a 3rd would appear if they later moved closer. – Hypnosifl Mar 17 '20 at 06:15
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    @Hypnosifl Perhaps, or perhaps they'll have no arithmetic or concepts, those are our devices. We or they decide what the rules are for them, if any, not the world. The question confuses linguistic conventions with what they are used to express. – Conifold Mar 17 '20 at 07:34
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    When you say 2+2=3, are you [using or mentioning](https://en.wikipedia.org/wiki/Use%E2%80%93mention_distinction)? Food for thought. – J D Mar 18 '20 at 03:58
  • Does this answer your question? [Can there be a universe with different mathematics?](https://philosophy.stackexchange.com/questions/9908/can-there-be-a-universe-with-different-mathematics) – viuser Mar 21 '20 at 03:59
  • https://mathworld.wolfram.com/PeanosAxioms.html Can an omnipotent being make Successor(Successor(0)) = Successor(Successor(Successor(0))) ? No. – polcott Apr 19 '20 at 15:33
  • No. Take a look at the points raised [here](https://philosophy.stackexchange.com/a/29350/14619). (Unless your definition of "3" is not the same as mine.) – user21820 Apr 23 '20 at 19:35

12 Answers12

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1 + 1 in exponential math can equal three, so this is not actually a good way to phrase the question. Some aspects of our world sum arithmetically, some sum exponentially, some sum logarithmically, some statistically, and some sum in the frequency domain. I think we all get the point, which is to ask about logical impossibilities, but the answer when one just considers the math actually suggests an answer.

We humans tend to think that what we experience on a daily basis is a logical necessity. In math terms, this was exemplified by the belief that Euclidean Geometry couldn't NOT be true -- everywhere and for everything. This was actually the example Kant used. Of course, less than a half century later, non-Euclidean geometries were developed that were self-consistent, demonstrating Kant was entirely wrong about the "logical necessity" of any form of math. And then our universe ended up following non-euclidean geometry, just rubbing the noses of necessitarians in their error. (Note that in a space of varying non-euclidean curvature, angles do not sum arithmetically either, for yet another example from our world.)

MATH, it turns out, can be done in all sorts of exotic and unique ways. Which of them are instantiated in the world -- is CONTINGENT, not necessary. And yes, a world could exist where exponential summing was common rather than an unusual exotica, and 1+1 = 3 on a pretty regular basis.

Your bigger question has to do with logic, and whether logic is necessary, even if math is not. And if logic IS necessary, would it then precede and constrain a creator deity? Alternatively, of a deity created everything, a PantoKrator, why would that everything not include logic?

Well -- it took a century and a half after showing that math is purely discretionary, but logicians are now pretty much agreed that LOGIC is also discretionary: https://math.vanderbilt.edu/schectex/logics/ If logic is contingent, then a creator, a PantoKrator, can in principle specify the logic that a world follows. And also could have different parts of a world follow different logics. Which we pretty much knew, given how quantum mechanics follows different logic than macro scale matter.

Dcleve
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    The problem is that when you refer to alternative systems of math you are changing the definitions of the terms (different axioms to define them)--for example if someone says it's logically impossible two parallel straight lines could ever meet, thinking about the intuitive meaning of the terms in Euclidean geometry, does it really prove them wrong if you show this isn't true when you redefine a "straight line" to mean an arbitrary geodesic on a curved surface, like a [great circle](https://en.wikipedia.org/wiki/Great_circle) on a sphere, and define "parallel' in terms of parallel transport? – Hypnosifl Mar 17 '20 at 04:41
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    @Hypnosifl -- I think I agree with you, I believe that a logic or math claim has to come with a specified frame set, as in "1+1=2 in linear arithmetic summing", and the selected frame set is contingent. – Dcleve Mar 17 '20 at 04:53
  • Kant didn't say that Euclidean geometry was "logically" necessary. – Kristian Berry Apr 22 '20 at 21:37
  • @KristianBerry -- this author disagrees with you: https://www.researchgate.net/publication/271880703_Kant_and_Newton_on_the_a_priori_necessity_of_geometry. and while this one agrees, he concedes that the view that "Kant does say that Euclidean Geometry is an a prior necessity" is a near-consensus: https://repository.hkbu.edu.hk/cgi/viewcontent.cgi?article=1016&context=rel_ja – Dcleve Apr 22 '20 at 22:05
  • Kant thought Euclidean geometry was necessary synthetically _a priori_, not analytically so. The latter would be "logical" necessity. – Kristian Berry Apr 22 '20 at 22:08
  • @KristianBerry This author disagrees, and uses both geometry and arithmetic as examples of analytic a priori in Kant: http://homepages.wmich.edu/~baldner/sap.pdf. And this one's definition of synthetic a-priori would not include them either. https://medium.com/@rgrydns/kant-how-is-a-synthetic-a-priori-judgment-possible-45af58688600 – Dcleve Apr 22 '20 at 22:36
  • Well in the first Critique itself Kant says that Euclidean geometry is known synthetically _a priori_ so I don't know what to tell you. – Kristian Berry Apr 22 '20 at 22:53
  • The two authors I linked point out that Kant was using the nature of "extent" and arithmetic summing to infer that arithmetic and geometry were logical prior assumptions embedded in our even considering matter and math. At the time, no non-euclidean geometry was imagined to be possible, hence Kant was arguing for a form of necessity for both arithmetic and euclidean geometry. His reasoning also presupposes the prior necessity of classical logic, and is a logical reasoning process. By inference, then, in Kant's language something "known synthetically a priori" IS "logically necessary". – Dcleve Apr 24 '20 at 15:48
  • @KristianBerry -- forgot to "at" you! Note, all three of the assumptions that Kant is cited as relying upon are now questioned by mathematicians and logicians. His paradigmatic cases of synthetic a priori, such that synthetic a priori produces a "necessity", are now rejected at NOT necessary. And, with the challenge to logic form and method being a discretionary choice, the possibility of ANYTHING being logically necessary is highly suspect. – Dcleve Apr 24 '20 at 16:39
  • Kant would have allowed that concepts of alternative geometries are logically possible. I suppose if you're using the word "logically" in some other sense, your remark goes through to that extent. – Kristian Berry Apr 24 '20 at 21:29
  • @KristianBerry -- the first two links I provided you show your interpretation of Kant is a small minority view among scholars. – Dcleve Apr 24 '20 at 22:04
  • Kant says that Euclidean geometry is necessarily true, but not analytically. He defines "logically" as "analytically" so that's that. – Kristian Berry Apr 24 '20 at 23:33
  • @KristianBerry -- Kant's claim of necessity is clearly false, the SAP link notes that Kant was using a different definition of "logic" than is used today. That his paradigmatic example of an unquestionably "necessary synthetic a priori" is not necessary or a priori makes him an excellent and noteworthy example of the false belief that our every-day experiences and presuppositions are "how things have to be". – Dcleve Apr 25 '20 at 16:11
  • Successor(0) + Successor(0) = Successor(Successor(0)) is an aspect of the set of stipulated relations between conceptual classes that define the semantic meaning of the knowledge of arithmetic. Brand new analytical knowledge is created by naming a new set of stipulated relations between conceptual classes. – polcott Jul 23 '20 at 21:17
  • @polcott -- this appears to be an irrelevant post with respect to my answer. – Dcleve Jul 23 '20 at 21:36
  • @Dcleve Your "proof" of the lack of logical necessity in math was anchored in the fallacy of equivocation error so I eliminated the equivocation. Successor(0) + Successor(0) = Successor(Successor(0)) is logically necessary in PA. – polcott Jul 23 '20 at 21:41
  • @polcott -- you have identified no equivocation, nor demonstrated any logical necessity. Nor tied anything to my answer. Your posts appear to be incoherent non sequiturs. – Dcleve Jul 23 '20 at 21:47
  • "it took a century and a half after showing that math is purely discretionary" Math is not purely discretionary. The concept of Successor(0) + Successor(0) = Successor(Successor(0)) remains immutable no matter how this concept is encoded in language. – polcott Jul 23 '20 at 23:40
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Can an omnipotent being make:
Successor(Successor(0)) = Successor(Successor(Successor(0)))? --No. https://mathworld.wolfram.com/PeanosAxioms.html

Can an omnipotent being assign the semantic meaning of {2} to the numeric symbol "3", --Yes.

polcott
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  • Just an attempt.

(1) If a proposition is contradictory, it logically implies any proposition.

(2) 1+1 = 3 is contradictory.

(3) God decides that 1+1 = 3.

(4) Therefore God decided that any proposition is true.

(5) Therefore, God decides that " 1+1 is not equal to 3 " is true.

(6) Therefore God decides that one and the same proposition has 2 truth values.

So , the question amounts to : " can an omnipotent being reject the principle of bivalence?".

  • Other attempt :

(1) An omnipotent being is a being that can do anything that is possible.

(2) It is not possible that 1+1 = 3.

(3) Hence, it is not the case that an omnipotent being can make 1+1 = 3.

  • Another attempt ( along Descartes' line of thought in Meditations):

(1) A bad or evil will is a sign of impotency.

(2) A world in which 1+1 = 3 is true is a bad world, for in that world if a proposition is true, it is also false, in such a way that truth and falsehood do not mean anything.

(3) God is omnipotent, infinitely powerful, hence infinitely good.

(4) God would never want to create a world in which 1+1 = 3, for this world would be a bad one.

(5) It is morally impossible God to make 1+1 = 3, because he is omnipotent.

  • thoughts: Your first (2), 2a, is not true for many forms of summing. Most entertainingly, it is not even true of people, as population math leads to 1+1 = 3, then 4, then many more. 2b presumes that logic preceeds a PantoKrator -- so, where did logic come from, and why can't it be modified? The multiple logics we have found, show that there is no single absolute "One True Logic". The argument in 6a unravels when one realizes that reasoning and logic cannot be justified by reason or logic -- the Munchausen Trilemma shows that reasoning cannot validly close on itself. – Dcleve May 02 '20 at 20:54
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If we're going to be inferring abilities from omnipotence, we'll be using some notion of inference, so we'll at least be holding the rules of that inference true even for our all-capable being. If 1+1=3 according to those rules, then...

Kristian Berry
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We first have to answer the question of what does omnipotence mean. If a being is omnipotent, is this being bound by the rules of logic, or can the being alter logical inference itself?

If God cannot alter the rules of logical inference, then if he created a universe where 1+1=3, this universe would automatically be degenerate. The rules of logical inference prove that 1+1=2 (e.g. use the Peano axioms). If 1+1=3 and 1+1=2 and logical inference is valid, then the principle of explosion implies that in this universe, every possible statement is both true and false.

However, if God can alter the rules of logical inference, then all bets are off. It is not possible for me or anyone else to rationally answer this question under that assumption, as doing so requires assuming rules of logical inference. I suppose one could argue that in such a universe, there would be nothing that we would recognize as numbers and so there wouldn't be such statements as 1+1=3, but since we have no way of conceptualizing such a universe, I don't feel comfortable to make any claim about it at all.

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You can do 1+1 apples only if both apples are absolutely identical. Can you find two apples having such characteristic? No way. What you will always do is selecting a scale to accept identity. That is "I will consider two apples if they weigh the same up to the limit of my balance, which shows hundredths of kgs". But in such case, you are just being subjective about what to consider an identity. You will be saying 1+1=2 to something similar to 1.0000+1.0001=2.0 (which is obviously wrong). 1+1=3 is exactly the same case:

Take a digital bathroom scale that's able only to display integers. Put 1.4 kg on it. It'll tell "1". Repeat with a similar weight. Now, put both on the balance, you'll read "3". That is, 1+1=3.

This is not a joke. A classical philosophical problem is the selection of a limit for the definition of objects. That means that what is 1 for you, could be 0 for another person. If you have a ton of apples in a truck and two persons count them, the results will ALWAYS be different, and not because of errors. Some apples will be partially smashed and one could count them as valid and the other might not.

In simple words, an object (represented mathematically by the unit, 1) is a subjective definition. The definitions of objects are never objective. Therefore, 1+1 is not factually possible, although we assume subjectively that it is, and accept that 1+1=3 in some cases. Therefore, 1+1=3 is a common fact, although we prefer to ignore it and focus the ideal provided by reason and the limitations of perception.

RodolfoAP
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  • Excellent point -- the logic of math NOT a constraint on our world, which does not have absolutes or categories intrinsic to it. – Dcleve Apr 22 '20 at 22:15
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All words and numbers are based on definitions on how we understand them. 1 Woman and 1 Man interacting with each other will possibly get you to a family of 3

So we are essentially already living in the world you described.

To give a little bit more context.

All Areas of Mathematics are based on Axioms. Those are non-proveable statements which are just assumed to be true and show themselve to be "at least somewhat correct" by providing Theories which are derived by them which in turn provide us with possibilities to check if everything works out by real life experiments.

For now everything in the Axiomatik world of Mathematics looks good. But no one can be 100% certain that it will stay that way.

So the direct answer to your question is "we can not know", but the most likely one is "yes, because 1+1=3 is already the case for some underlying Definitions"

  • Excellent point, population dynamics is exponential, not arithmetic, so sometimes 1+1 does equal 3. – Dcleve Apr 22 '20 at 22:16
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It depends on the definition. For instance, if you have one infinity and add another infinity, is it now 2 infinities? or is it just a greater infinity? This theory could also be reversed. Making 1 infinity + 1 infinity = 3 infinities.

Foolish Lemon
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  • https://mathworld.wolfram.com/PeanosAxioms.html Can an omnipotent being make Successor(Successor(0)) = Successor(Successor(Successor(0))) ? No. – polcott Apr 19 '20 at 15:32
  • Good point, arithmetic summing does not work with infinities. However, fleshing the answer out with more content, so that it stands alone as an answer, rather than just a comment, would be a good idea. – Dcleve Apr 22 '20 at 22:12
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The most dramatic example I could think of is 1+1=3. Ex: One shoe, another shoe, bam - 3 shoes. This concept of a world where 1+1=3 is really hard to conceptualize for me because it changes everything that I understand the world *

I am not quite sure what you mean by "One shoe, another shoe, bam - 3 shoes". Are you saying a world where law of conservation doesn't hold as it does in our World? In our World, you can make 3 shoes out of 1+1 (make smaller shoe -2/3rd of original); you can now count them up to 3, and probably even say, falsely though, 1+1=3.

If the other World is different, you may end up having 1+1=3 because that is how that World would be like. 1+1=3 will be the norm, and your question, under its current interpretation, will have no value.

Ajax
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  • Excellent point. Our world does not have "conservation of entity quantity", hence 1+1 can equal 3 in all sorts of situations. – Dcleve Jul 23 '20 at 21:08
  • @Dcleve Yes, and it suggests that Type Theory can be a (reasonable) foundation of mathematics. – Ajax Jul 24 '20 at 17:48
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When we say that p is necessary we mean that it is not possible that not p . So when we say 1 + 1 = 2 is necessary we mean that it is not possible that 1 + 1 is not 2 . To say that God is able to make 1 + 1 = 3 is to say that God is able to do something that is impossible .

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How do you mean that? In the cause of maths it can be done, by simply omitting the alignment of the symbols, but math would still work the same. If you mean, it as in lying, there the ground rule of a world is to grift about The Truth and telling it is considered wrong and undesired, it could also be. I just don't know, why you would want that. The Romans had something similar, then it was considered normal to offer briber in court and have mobs intervene, being the fundamental part in it, also slander as a form of argument to further the plot. Obviously, in the course of history, it is no longer a common practice.

Walgekaaren
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A world in which counting is meaningless is necessarily a world in which measurements are meaningless. So if in that world 1+1 = 3, then the distance to the sun is both 93 million miles and zero. Such a world would be uninhabitable to humans.

niels nielsen
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  • Maybe if 1+1=3 were true, then all living things would look different, as well. Such a world might be inhabitable to some other human-like animal. – Mark Andrews Mar 17 '20 at 00:52
  • Your premise is quite simply false. Logarithmic, exponential, and statistical summing are not arithmetic, and some aspects of the world use all of them. We currently inhabit a world where 1+1 does not always equal two. We DO tend to require reasonable stability in our world, but either absolute stability, or arithmetic summing -- no. – Dcleve Mar 17 '20 at 03:38
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    The OP's question did not specify what form of summing to consider. I took the simplest example. If we accept 1+1 = 3, subtracting 2 from each side yields 0 = 1, an important result which establishes among other things that Winston Churchill is a carrot. Proof is left as an exercise to the reader. . – niels nielsen Mar 17 '20 at 04:53
  • Subtracting 2 from each side of 1+1=3 in exponential math does not yield 0 = 1. Your "proof" has presuppositions that are false. Have you ever done statistical summing or subtractions. Logarithmic, or frequency summing or subtractions? – Dcleve Mar 17 '20 at 04:57
  • I stated my presuppositions, they are not false, yes I have, but I will waste no further time on this. Since this is "philosophy", you are free to believe whatever you want. – niels nielsen Mar 17 '20 at 07:00