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2 + 2 = 4, right? If I have 2 things, and you have 2 things, we have 4 things between us. This is a logical reality that is inescapable. If we want a coherent, consistent set of rules for counting, this is the simplest and most useful, as well as what applies to the created physical (and spiritual?) reality we are in. The fact that this is the most useful, coherent and consistent system of counting is a "set in stone" fact that belongs to the Platonic world.

Question: Defining the Platonic world as Sir Roger Penrose does in "The Road to Reality"; the "perfect" world of logical structures (logic, mathematics), what exactly is Hashem's relationship to it? Is it any of the following (or something else?):

1. He created the Platonic world, and designed it.

He decided that 2 + 2 = 4 in the above counting system. He could have made it so that 2 + 2 = 5, without modifying the meaning of "2" or "+" or "=" or "4" or "5" (i.e. keeping all of that exactly the same), Hashem could have still, in His omnipotence/lack-of-definition said "Let 2 plus 2 be equal to 5", and so it would have verily been and we wouldn't have been able to fathom 2 + 2 = 4 in the same way. We can't imagine how that would work, but we are compelled to believe this in light of our theology.

2. He created the Platonic world, but one does not "design" the Platonic World.

It's not possible that 2 + 2 = 5 (again, without the obvious modification of "2", or "+" or "5", or switching counting system, which is just a convention, and I am trying to be fundamental here). There's no possibility of a universe where, if you have 2 things, and I have 2 things, together we have 5 things and that makes sense. Asking Hashem to do that is as meaningless as asking Him to create a square circle. However, there was once a time when there was absolutely no such thing as squares, circles or "2" or "5" or "+", not b'po'el, not b'co'ach, not b'svara. Hashem created all that in the same way He created anything else.

3. One does not "create" the Platonic world, Hashem is also bound by its truths (ch'v)

Given that 2 + 2 = 4 and that's inevitable, then Hashem didn't create that. That's just the way it is. Hashem just "enabled" it, by creating in general, He unlocked possible, but He didn't need to do anything in terms of what's possible, that just is. 2 + 2 was always going to equal 4, and Hashem has always known that. I say chas veshalom because just expressing the idea that Hashem is "bound" by something sounds very wrong, which is why this position, while seemingly inescapable to my small, finite mind, is going to take the most work to prove.


Couple of extra points:

  • I don't want to get too into the philosophy of Plato. I am just borrowing his name to quote the idea of this perfect world of logical and mathematical forms in the most loose sense so we can talk about it. Anything that he said about it isn't necessarily what I am holding. E.g. we don't need to discuss here if the Platonic world is "real" or not etc. I think that's a bit much for this question and should only be addressed if absolutely necessary. My question is what do Chazal and Torah say on this, not him or any other secular philosopher (although they can be brought for context or utility!).
  • I have been asking this question for a long time. I've only ever had one answer and that is (1) above. Rabbi Dovid Gottlieb told me that "yes, that's what it seems the Torah is saying, but it's very hard to understand". I've also seen it come up many times in Chabad philosophical writings, in fact it's quite a key idea behind a lot of Kabbalistic points (such as the idea that Hashem created principles, including the principle of existence itself - which is a Platonic structure)
  • I don't feel qualified to "accept" an answer here but it will fall on me. I do apologise for any bias I have. I have spent most of my life believing (3) and expecting a good explanation of how it doesn't detract from Hashem's omnipotence, but lately, I think my bias is towards (1). I will look at votes, but the ideal answer will have a good explanation, go into as many details as possible (for example, if 1 is being brought, one should try to explain to some extent how it is possible that He could have made 2 + 2 = 5, and if 3 is being brought, some explanation on how we can understand Hashem being bound by anything etc), and have strong sources.
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  • 1
    sacred-texts.com/jud/gfp/gfp151.htm
    – Alex
    Jan 15, 2023 at 16:31
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    The world of mathematics is a human invention, designed to help describe, use and possibly predict the real world. I recommend The Fabric of Reality by Professor David Deutsch. Jan 15, 2023 at 16:55
  • @Alex thanks +1 for MN. Would you say that Rambam is saying that Hashem neither created, nor designed the Platonic world and holds by (3)? I'm not convinced he is saying enough here to deduce what he holds.
    – Rabbi Kaii
    Jan 15, 2023 at 16:59
  • @MauriceMizrahi thanks Dr. is that a Torah source? I believe what you have just said is contentious among philosophers. I recently had an encounter with a physicists who told me, in almost pain, that after 50 years of physics, he's convinced that the bottom rung of reality is mathematics...
    – Rabbi Kaii
    Jan 15, 2023 at 17:01
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    Personally I think (2) makes the most sense. Somewhere in Emunos v'Deos Rav Saadya Gaon says people ask if Hashem could fit the Earth through a regular ring without changing the size of either object. He says it is a silly question. While it is true Hashem can do everything, logical impossibilities do not count as things. Especially in regards to 2+2 equalling five, the definition of 2 and 5 are related. You can't make 2 plus 2 equal five without changing their definition. It is like asking, "What is the sound of one hand clapping?" The answer is by definition it's not clapping.
    – N.T.
    Jan 17, 2023 at 10:06

3 Answers 3

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First, we would have to establish the Platonic world exists. Aristotle said it didn't. This particular 1 apple and 1 apple making two apples exists, but the general pattern is a rule people extracted, and doesn't have an existence.

The Rambam actually uses this point when speaking of the concept of "human being". Individual people exist, but there is no created concept of "human being" in general. He writes (Moreh Nevuchim 3:18m in Freodlander's 1903 imperfect but public domain translation):

It is an established fact that species have no existence except in our own minds. Species and other classes are merely ideas formed in our minds, whilst everything in real existence is an individual object, or an aggregate of individual objects. This being granted, it must further be admitted that the result of the existing Divine influence, that reaches mankind through the human intellect, is identical with individual intellects really in existence, with which, e.g., Zeiḍ, Amr, Kaled and Bekr, are endowed.

Different people fit the category more or less, because that's all it is, a generalization people's minds constructed from the various instances they encounter.

But when it comes to logic, the Rambam considers it a facet of Truth, and thus of the Divine Essence. From 3:15:

THAT which is impossible has a permanent and constant property, which is not the result of some agent, and cannot in any way change, and consequently we do not ascribe to God the power of doing what is impossible. No thinking man denies the truth of this maxim; none ignore it, but such as have no idea of Logic... Likewise it is impossible that God should produce a being like Himself, or annihilate, corporify, or change Himself.... Again, whilst philosophers say that it is impossible to produce a square with a diagonal equal to one of the sides, or a solid angle that includes four right angles, or similar things, it is thought possible by some persons who are ignorant of mathematics, and who only know the words of these propositions, but have no idea of that which is expressed by them. I wonder whether this gate of research is open, so that all may freely enter, and whilst one imagines a thing and considers it possible, another is at liberty to assert that such a thing is impossible by its very nature; or whether the gate is closed and guarded by certain rules, so that we are able to decide with certainty whether a thing is physically impossible.

That chapter has more problems with figuring out what is logically impossible than with saying Hashem cannot do a paradox.

But either way, it is no different to the Rambam whether one asks why Hashem couldn't make 2+2=5 than why He couldn't catch a cold, make a fatzlapah or any other meaningless combination of syllables. The thing you're asking for makes no sense.

So, the Rambam picks other options to your #5. Either that illogical systems are meaningless mouthings, or that Hashem isn't so much "bound by" logic as He is Truth, and logic is a consequence of His Esssence.


But there are plenty of Rishonim who are more Platonist than that, so it would pay explore the question from their perspective...

The Ramchal says that logic itself is a nivra, a created "thing", and therefore Hashem couldn't be subject to it. So, we could have lived in a universe where 2+2=5, but it wasn't His Will to be so. Humanity needs a logical and consistent system of logic, or else we cannot reason about anything, and Free Will would be meaningless.

So Hashem set up some system of logic and math because it fit His "Desire" to provide us with meaningful Free Will. Now the question is why this system and not that.

The past century plus has shown that there are multiple possible logics — multivalent logic (values other than true and false exist), fuzzy logic, Quantum Logic (in which conflicting possibilities coexist until the "wave function collapses"), and many others. Logic is no longer one system that must be true in any context. Which argues in favor of the Ramchal’s position, that these logics are created and not an aspect of Divine Truth.

Also, according to Noether’s [1st] Theorem, every differentiable symmetry of the action of a physical system has a corresponding conservation law. In other words, energy is conserved because physics is symmetric over time, ie the laws of physics don’t change. Momentum is concerved because physics doesn’t change based on location, angular momentum is conserved as an aspect of the laws of physics being the same in every direction (even if in our situation, gravity isn’t the same), etc…

We are getting very close to the point of finding that logic underlies physical law. In which case, we would be forced to conclude that any miracle that defies nature would require Hashem defying logic.

So, I think that because of secular developments that post-dated both, the existence of "the Quantum Realm" in which Aristotelian logic fails and the existence of miracles in a post-Noethe understanding of the universe both point to logic being a nivra, and not a feature inherent to Truth.


You might also want to see this discussion of a related question, the Euthyphro Dilemma. In a monotheistic form, the question is whether Hashem Decided what Good and Evil are, in which case there is no cause for Good to be Good, and it's all artbirary, or whether Good and Evil are pre-existing criteria, and thus implying that Hashem was subject to pre-existing ideas of good and evil. My answer there: https://judaism.stackexchange.com/a/118147/1570

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  • Still contemplating all this, thanks for the journey. I would accept this answer right now but I want to encourage more answers.
    – Rabbi Kaii
    Jul 28, 2023 at 12:47
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I was almost inclined to vote to close this question because it doesn’t really seem to be about Judaism on the surface.

But to give you the benefit of the doubt and reading carefully what you wrote, it seems that what you are asking about is really pertaining to the subject of the Sefirot discussed in kabbalistic literature.

What you are calling the Platonic World, you define in terms of logic and mathematics, meaning quantification.

In the nomenclature and context of Torah you are describing the Sefirot (ספירות). When trying to understand the concept of Sefirot a good analogy is to think of the distinction between G-d and His name. He and His name are one and yet He, at His essence and being, transcend even His name. His name is only a vessel or tool that enables us to connect.

This is in keeping with what is found in the first chapter of Sefer Yetzirah and the 3 variations of the root ספר, ספר,ספר.

The first of these 3 terms is referring to the general category of quantification, meaning mathematics and logic in the broadest sense. A related idea would be the mathematical expression of ciphering. This category also pertains to the beginning of the introduction of measured or linear time (זמן משוער) and its transition from the earlier state of non-linear time (זמן בלתי המשוער). This is also called the general category of שנה.

The second of these 3 terms is referring to the general category of form. Again in the broadest sense. Linguistically associated with Sofrut and the formation of letters. This category can be associated with Space or Place and is also called the general category of עולם.

The third of the 3 terms pertains to discrete being as expressed through the 5 Gevurot. Like it explains in Sefer Yetzirah, this is understood by way of example through the function of speech and the formation of words by the 5 parts of the mouth which divide and break the simple sound or voice into speech. This is the idea of Sippur, telling. This final category is also described as Soul נפש, which is similar divided into 5 terms (נפש, רוח, נשמה, חיה, יחידה.

This group (עולם, שנה, נפש) is what is alluded to in the Torah when it says Moshe Rabbeinu entered the cloud (עש״ן) when ascending Mount Sinai to receive the Torah from G-d.

This Sefirotic system is essentially what makes creation and existence as we understand it, to be. And this is in contrast to G-d at the level of Being or Essence, which transcends any expression and is unknowable.

If you are interested in a very detailed explanation of the concept, there is a Chassidic discourse from the previous Lubavitcher Rebbe, Rabbi Yosef Yitzchok in his Sefer HaMa’amarim from the year 5685 for parshat Shemot which begins with the words:

הבאים ישרש יעקב יציץ ופרח ישראל כו׳

You can find this volume at Hebrewbooks.org.

Enjoy the Ma’amar!

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  • Thanks, I appreciate the reference, I look forward. So to sum up then, which of the 3 above positions do you think Kabballah holds (or is it none of them?). Thanks for not voting to close the question, I am looking for Torah based answers only on this question of Hashem's creating, so I am not sure what would be inappropriate about it for the site?
    – Rabbi Kaii
    Jan 15, 2023 at 18:00
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    @RabbiKaii No.1 for sure, but the part of No.3 that says G-d is bound by the system is also true. This is the principle of מגיד דבריו ליעקב ומשפטיו לישראל. The emphasis being that they apply to G-d also. This concept is also discussed in Talmud with such ideas as G-d putting on tefillin, etc. Jan 15, 2023 at 20:25
  • This answer gets deeper each time I read it.
    – Rabbi Kaii
    Jul 28, 2023 at 12:53
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I recently stumbled across this passage of שער האמונה ויסוד החסידות, in פתח השער לבית יעקב ד׳, which unequivocally supports #1:

אכן צריך האדם להאמין באמונה שלימה, שהשי"ת יכול לשנות באדם זאת ההשגה והשכל, להשגה אחרת ושכל אחר, בין בכוללות ההשגה לשנות כל סדר המערכה במעשה, שישנה שיתבטל כח הבחירה מהאדם, ובין שישנה רק ההשגה והשכל, וכח הבחירה לא יתבטל, ויכול ליתן באדם שכל אחר והשגה אחרת, שיבין וישכיל בכל דבר היפוך ממה שמבין עתה. וזה דאיתא במס' (תמיד כח.) איזוהי הדרך שיבור לו האדם וכו' יחזיק באמונה יתירה שנאמר (תהלים קא) עיני בנאמני ארץ לשבת עמדי, והוא שיאמין באמונה שלימה שאין שום דבר נמנע מהבורא, וגם דעת האדם הוא תמיד ביד הש"י. שכמו שאנו מבינים בשכל הזה ובידיעתנו ובהשגתנו הזאת, שאי אפשר לשני הפכים להיות בנושא אחד, כן יכול לחדש לנו דיעה ושכל אחר והשגה אחרת, שנשכיל ונבין ונדע בידיעה גמורה שאפשר להיות שני הפכים בנושא אחד. וכדאיתא בזה"ק פ' בא (דף מב:) הנ"ל דאנא יכיל למחאה ההיא צורה ולמעבד לה כמה זמנין ולית אלוה אחרא דיכיל לממחי וכו'. וכמ"ש לעיל, שמאחר שהשגתנו ושכלנו וידיעתנו הם נבראים ונסדרים ונערכים, ע"פ רצון הש"י מה שחלק לנו וסדר לנו, ויכול לשנות ולהפך מהפך אל היפך. וכמו שנבין ונדע עתה בירור גמור ששנים הם יותר מאחד, כן יכול השם יתברך להפוך שכלנו ודעתנו שנבין בבירור גמור, שאחד הוא יותר משנים:

Indeed, a human being is enjoined to believe with perfect faith that God can change his understanding and power of intellect to a completely different one, including one’s overall intellect, that is, the entire order and structure of his mind. This could mean that man’s power of free choice could become completely nullified, or it could mean that only his understanding and power of intellect is changed, yet his free choice remains intact. It could be that God gives man a completely new way of understanding totally different from that which he now knows. This is as it is said in the Gemara (Tamid, 28a), “Which is the right way that man should choose? Let him have abundant faithfulness, as it is written (Tehillim, 101:6), ‘My eyes are upon the faithful of the land that they may dwell with Me.’” This means to believe with perfect faith that nothing is impossible for the Creator, and similarly, man’s mind is in God’s hands. For example, just as it we now understand with clarity of intellect that it is impossible that two opposites could be true on any one subject, it is in God’s power to grant us a perfectly clear understanding that two opposites could be true in one situation. This is as it is written in the Zohar (Bo, 42b), quoted above: I can erase its form and re-create it as many times as I want, and there is no god above Me that can erase My “form.” As was mentioned previously, since human knowledge and understanding is a created entity, ordered, arranged, and granted to us according to God’s will and desire, it could be God’s will and desire to change it to its very opposite. Just as we know now with perfect clarity that two is more than one, so too is it possible that God can have us believe with complete clarity that one is more than two.

He then brings from R. Yedaya Ben Avraham that the Rashba would teach position #2, but that this is wrong, and then proceeds to bring proofs from various sources, such as the Ark not taking up any space in the Holy of Holies, or the people bowing with enough room even though it was packed in the Beit Hamikdash, and how really contradiction and impossibility are only constructs of the way our intellect has been created and programmed by Hashem, and when He desires, He can change it and we have no issue understanding that these "impossible" things are just as acceptable as anything else. Indeed, one day we will no longer be restricted like this.

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