I realize this is a very common question, but the problem of evil is an important one, and as far as I can tell, has not been asked here.
In short, why does G-d allow evil to exist?
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I would say that an answer on the simplest level would be to give us בחירה - choice. If there was no evil (and no "evil inclination"), there would not be much meaning to doing "good" and fulfilling the Torah. If everything was good and clear-cut, we would not really have free choice.
On a deeper level, this question could probably fill books (and likely does)
Evil exists because Gd desires for it to exist. Just as gravity exists, planets, entropy, time, space, animals etc. Everything in the world has it's purpose and one can find positive reasons for all things, and negative consequences of all things. For example, if gravity didn't exist, people would not be able to stay on the planet, however because of gravity, people get hurt when they run and fall, and bridge construction causes people to die.
Evil, has its own positive consequences as well.
The Ramchal in Da'as Tevunos gives several reasons for the creation of evil:
In siman 38, he explains that Hashem's Oneness (the revelation of which is the purpose of creation - siman 40) can only be fully grasped by seeing the facade of what it would mean for Hashem to not be in complete control, and then negating that facade. In siman 40 he writes that Hashem made evil in order to use it as a vehicle for revealing His complete absoluteness.
הרע שברא הוא ית' להודיע אמתת יחודו ולנסות בו את האדם
...the evil that He created in order to make known His true Oneness and to test man with it
I briefly discussed how this works here.
In this point, the Ramchal is arguing that the primary purpose of the existence of evil is not for the purpose of facilitating free-will, but rather for the purpose of completing the appreciation of Hashem's Oneness. Evil, and the notion that there is a reality to something outside of Hashem's will, exist only to facilitate the revelation of Hashem's Oneness when evil is obliterated.
Later (Siman 118), discussing the existence of bad things, e.g. suffering, he says evil was created in order to have a mechanism for punishing the wicked:
ונמצא כלל הרע, אשר הוא ית' השיב אחור ימינו מהנהגת עולמו, וינהגהו בכבדות, תוקף חשך סתרו, ושם בטבע הלז כל הפסדי הטוב, הלא המה כל מדות בית דין, להביא במשפט על כל נעלם ורשעים
(Paraphrased) Evil was created to bring punishment upon the evildoers.
Evil in this context means the existence of negative things. If Hashem is completely good and bad is an "external" creation (siman 104), the existence of negativity needs to be explained.
As for the Evil Inclination, Da'as Tevunos cites the Zohar (Siman 78):
מאמרם ז"ל (זוהר ח"א, קו ע"ב), לא נברא יצה"ר אלא לנסות בו את בני האדם"
The Evil Inclination was only created to test man.
G-d created good and evil, says both the Torah and Isaiah (Deut. 30:15-20; Isa. 45:7). One reason for having both is explained at Deut. 30:15-20 -- with Free Will, G-d had to give us choices, both good and bad, and the inclination to pursue evil as well as the competing inclination to pursue good. Without these inclinations fighting each other, our election of one over the other would be a foregone conclusion just like the last Egyptian election. Secondly, he created other kinds of evil -- natural disasters, still-born babies, birth defects, diseases, poor people -- to teach us His mida of compassion. Without a communal need that people give of themselves, we would lack community.
Of course, good and evil in this world are completely relative. What is good for the butcher is an evil for the animal to be slaughtered. We see this very much in the institution of karbonot (sacrifices) of which the Torah spends much time discussing even though animal offerings have only been an active part of our religion for a relatively short period (about 1400 years). As I heard brought down from Rabbi Shmuel Goldin, we brought animal offerings for unintentional or unwitnessed sins in order to appreciate that but for certain factors, our own fate might be the same as this poor animal. It is ironic that we should learn compassion through the suffering of others, but how else would we learn it?
For more thoughts on this, see Rabbi Benjamin Blech's book, "If G-d is Good, Why is the World So Bad?", Simcha Press 2003.
The Tanya at the end of Chapter 9 bring a parable from the zohar illustrating the role of evil.
אך נפש הבהמית שמהקליפה, רצונה להפך ממש
But the animal soul derived from kelipah desires the very opposite; it desires that the body be pervaded with its faculties and its thought, speech and action.
לטובת האדם, שיתגבר עליה וינצחנה, כמשל הזונה שבזהר הקדוש
But the animal soul desires this for man’s benefit, in order that he prevail over her and vanquish her, as in the parable of the harlot [related] in the holy Zohar.19
The parable: A king desired to test the moral strength of his only son. He had a most charming and clever woman brought before him. Explaining to her the purpose of the test, he ordered her to exert every effort to seduce the crown prince. For the test to be valid, the supposed harlot had to use all her charms and guile, without betraying her mission in the slightest way. Any imperfection on her part would mean disobedience, and the failure of her mission. However, while she uses all her seductive powers, she inwardly desires that the prince should not succumb to them.
So too in our case: The kelipah itself desires that man overcome it and not permit himself to be led astray. The entire stratagem is solely for man’s benefit.
This is a good question and one that should perhaps be reviewed from time to time in ones life as knowledge grows and perspective matures.
To restate your question: "Why does evil exist and why does G-d allow evil to exist?"
Evil exists, like with everything, because G-d created it, like is understood from the Yotzer prayer said every Shabbat morning before reciting the Shema.
ברוך אתה ה' אלהינו מלך העולם. יוצר אור ובורא חשך. עשה שלום ובורא את הכל
So the first part of your question is dealing with the origin of evil according to Jewish tradition. That pertains to the subject of explaining the process of Creation.
Why G-d allows evil, the second part of your question, is revolving around the subject of the purpose for Creation, meaning what is G-d's purpose for the creation of everything.
Evil in Hebrew is translated as רע.
Reflecting on this for a moment and considering its deeper meaning, while contrasting and comparing it to its antonym will help to answer your questions.
With Holy studies in Hebrew, we are taught to observe from different hermeneutical perspectives. With a two letter root like רע that means from beginning to end and from end to beginning. This is in keeping with a general Torah principle set forth in Sefer Yetzirah 1:7 that the end is wedged in the beginning, and the beginning is wedged in the end.
עשר ספירות בלי מה נעוץ סופן בתחלתן ותחלתן בסופן כשלהבת קשורה בגחלת שאדון יחיד ואין לו שני ולפני אחד מה אתה סופר:
Looking from beginning to end, רע means evil. From end to beginning, as in ער, it has a connotation of arousal עיררות from the root ערר.
Following the same pattern of doubling the trailing letter (How the ער root is built into the 3 letter root ערר.) to build the three letter root, רע would become רעע which means to shatter or to break apart. This relates to a principle found in kabbalistic literature called the shattering of the vessels (שבירת הכלים) at the beginning of Creation. And on a higher level, this corresponds to the shattering or breaking apart, so to speak, of G-d's oneness as we perceive it, like is explained in Tzidkat HaTzadik 206:1 discussing the concept of G-d manifesting His Kingship where it says:
ושורש השרשים הוא הגיאות כנודע דשורש הרע הוא מעולם התוהו ושבירת הכלים הבא ע"י מחשבת אנא אמלוך כידוע
It is worth noting in the language of the text quoted here that the term הגיאות, which is called the root of evil רע, has an unusual spelling. Its correct spelling is הגאות, like in Devarim 33:26 and Tehillim 10:2, 68:35, 89:10 and 93:1.
Like is known, Hebrew word roots possess two, three or four letters normally. When extra letters are seen, it is to emphasize a special teaching and possibly a compound word derived from two or more words. הגיאות, which possesses an extra letter Yud, (and is a form not found in the entire Tanach) has multiple meanings. One being haughtiness or pride, meaning self-important and the other being majesty which is a term referring to the supreme dominion and authority of a Monarch מלך.
But with the additional letter Yud, it suggests a compound of two or three elements גיא-אות (Depth-Letter) or גיא-צלמות (Valley of the Shadow of Death). That the letters comprising G-d's name (and letters in general) introduce the concept of descent from Above to Below, even to the opposite of life. And this descent from Above is emphasized even more with the term צלמות which is itself a compound of G-d's Supernal name צלמו, אל עליון and the letter Tav which is associated with both life and death. In the complete compound גיא-צלם-אות, the form צלם indicates when the name of G-d associated with Feminine צלמו is diminished from 4 letters to 3 which establishes the paradigm which results in the reduction of one of the two great and equal luminaries to become the Moon (the concept of Giver and Receiver) on the fourth day of Creation, like is found in Bereshit 1:16. This paradigm also continues with the Creation of Adam HaRishon and the separation into Husband and Wife. This is truly the Root of all Roots like the Tzidkat HaTzadik writes.
And this is rooted in the Torah principle that there is no King without a people like is found, for example, in the writings of Rabbeinu Bachya ben Asher, Kad HaKemach, section Rosh HaShanah 3.
כי אין מלך בלא עם
That in order to manifest as King meaning מחשבת אנא אמלוך, the King must be chosen by the people. The King needs His people in order to be King, so to speak. Assuming the position by force of will is called a Governor מושל, not a King.
The antonym or opposite of evil רע is good טוב.
One of the deeper insights about the meaning of good relates to its numerical or gematria value.
We learn at the beginning of the first chapter of Sefer Yetzirah, a book attributed to the Patriarch Avraham, that Hebrew language is interpreted and understood according to three general hermeneutical principles in addition to other analytical methods.
Those three general principles are: quantification as in cypher ספר, meaning gematria and notarikon, form or orientation in space, like in Sophrut סופר which deals with the form and shape of the letters, and pronunciation (which involves both the simple sound of the voice and breaking that sound into the various types of letter combinations that form words via the mouth) as in to tell or to say the letters and words ספור.
It should also be noted here that in the language of the Sefer Yetzirah mentioned above discussing the relationship of beginning and end, the same word root used to categorize the hermeneutical categories for Hebrew letters ספר is also used to describe the relationship of beginning to end, namely ספירות.
And additionally, this paradigm for differentiation and categorization (of Hebrew letters and everything else ) is compared to a flame connected to a burning coal.
That G-d יהוה is called a Consuming Fire in Devarim 4:24 and 9:3.
It emphasizes that the flame and burning coal are one, whose base, (Like the silver bases or sockets used in the construction of the Mishkan. See Shemot 26:19, 21, 25 and elsewhere.) אדון (which is also related to אדני,G-d’s name associated with Kingship) is one and not two. In other words, G-d and (the letters which comprise) His name are one and not two.
טוב (good) has a numerical value of 17. In ascending numerical sequence, this is one away from and what leads to the next number in the sequence, meaning 18. 18 is חי, and has a connotation of life.
This is like we find in BeMidbar 14:28, Isaiah 49:18, Jeremiah 22:24 and Ezekiel 5:11 which says, "I am life, says G-d."
This adds meaning to the concept of good in its relational sense as it is intended in Torah, that good is what leads to an increase and diversification of the expression of life חי, like in פרו ורבו (see Bereshit 1:22-31), the commandment to Be fruitful and multiply. That the Small Sum gematria מספר קטן of פרו with the kollel is 17, and so too the Small Sum gematria of ורבו with the kollel. Two times טוב or טוב מאד.
It is important to distinguish that חי life here means the actual life-force or essence, not lifestyle, which would be סגנון (which hints [ס״ג and נו״ן] to something rooted in judgement and severity) in Hebrew or culture, which would be תרבות in Hebrew and can also allude to severity as in ריב quarreling.
The increase and diversification is of the actual life-force which means reproduction of the species (פרו ורבו) like in Bereshit 1:22, which is a part of the nature of all living things.
That good טוב, meaning the increase of life, applies throughout the entire universe and is also implied from the numerical value of good טוב itself, which is 17 like was said above.
This value of 17 also happens to be the numerical value of the Notarikon from the phrase in the first sentence in the Torah, "the Heavens and the Earth"(את השמים ואת הארץ):
(בראשית ברא אלהים את השמים ואת הארץ)
The Notarikon of this phrase, א-ה-ו-ה is understood to mean the whole of Creation, time שנה (non-linear time which existed prior to the fourth day of Creation and linear time which began on the fourth day like is found in Bereshit 1:14) and space מקום and the third aspect of creation which is called Soul נפש, the entire universe העולם. It is also one of the names of G-d and is what is intended, for example when referring to the Ba'al Shem Tov הבעל שם טוב and also the Good Name שם טוב mentioned in Avot 2:4, 4:13 and the Good טוב mentioned at the end of Avot 6:3-4.
And regarding this concept of עולם, שנה, נפש which is the Notarikon עשן, see Bereshit 15:17-18 concerning the Covenant between the Halves, and Shemot 19:18 and 20:14 and the Giving of the Torah at Mount Sinai. And like is taught by Rabbi Chaim Yosef David Azulai in Sefer Midbar Kedmot, Ma’arechet Dalet, 4:19, עשן also stands for עטיו של נחש Fangs of the Serpent, which he shows is a reference to Moshiach (See also tractate Shabbat 55a and tractate Bava Bathra 17a).
So the whole of Creation is intended to reveal G-d's presence חי, that G-d is one, in ever increasing ways throughout all of the universe טוב, meaning within the diversity.
Creation, therefore, has intended purpose תכלית, which is good, the increase and diversification of life, everywhere.
Evil רע is the opposite. It is pointless arousal or senseless passion and without purpose (similar to pointless hatred שנאת חנם, the cause of the last and longest period of exile) in the sense of an impulse, meaning a sudden strong and unreflective urge or desire to act. It results in the decrease of life and the absence of life's diversity and profusion. This impulse is only a flash of thought without reflection. But as the Torah teaches, G-d's thoughts (even if only on the level of impulse) are like actions from our perspective. This is one of the meanings related to My Thoughts are not like your thoughts (אין מחשבתי כמחשבותיך) and how G-d connects thoughts of Good (including His thoughts, which are good only) to action. (מחשבה טובה הקב״ה מצרפה למעשה)
It (evil) conceals G-d's oneness and presence in all things, leaving an appearance of separateness and disconnected existence. This is the concept of the Hebrew letters which make up G-d's name and everything within Creation.
Some suggest that Evil may somehow be associated with something outside of G-d's will. But as the Torah teaches in Jeremiah 23:24 explicitly, there is no place (meaning within the entire created universe) devoid of Him (including evil).
הלוא את־השמים ואת־הארץ אני מלא נאם־יהוה
And this raises the question, "From where in the process of creation (within G-d's will) does Evil arise?" Is it an essential part of G-d's Creation or something not wanted, nor intended, that we brought about afterward contrary to G-d’s will? This is the foundation of your original question.
And the answer to this is also alluded to above in regard to the subject of the shattering of the vessels, but only after careful consideration of the Hebrew language used.
That like there is beginning/first (תחילה/ראשית) and end/last (סוף/אחרון) in regard to Creation, there is also an inner פנימי and an outer חצוני aspect to all of Creation, like is taught by Rabbi Moshe Chaim Luzzato in Sefer Adir B'Marom, page 92, beginning with the words,
הנה בתחלה סידר המאציל את הבריאה בסוד פנימיות וחצוניות
Meaning Essential/Primordial Being, which is called מהות, and environment, called סביבה or מקיף in Hebrew. And there is the point of interface between them called עצמות which can be thought of as Essential/Primordial Form and Structure or מערכת אלהות. This paradigm of form and structure is the concept of the Sephirot mentioned above from Sefer Yetzirah as סופר.
This is somewhat analogous to ones physical self and the environment in which one resides or dwells and the skeleton which provides structure to the physical self.
Within kabbalistic and chassidic literature, this external, interfacing aspect is often compared to the ideas of vessels כלים or garments מלבושים. They encloth לובש or contain מקבל the inner wearer or contents מהות, provide boundary and form and serve as the interface (ממוצע המחבר or רקיע) with the even more distant external environment סביבה. If these garments are made and used properly, they fit closely to the qualities of the wearer (מהות) and serve their intended purpose, allowing positive interaction between the essence within and the environment without.
This is like when a soul is dressed in a healthy physical body. One is called alive, being alive similar to the expression, the Living G-d אלהים חיים like in Jeremiah 10:10.
And this external, interfacing aspect also pertains to the Torah concept of names (that the Torah itself, all the letters contained within it, is called the name of G-d). It should be noted that a name עצמות is the vessel or tool by which another סביבה (meaning the Creation) can grasp the name's possessor מהות, the Essence or being within.
They, meaning the name עצמות and the one to whom it applies מהות, are not identical. The name, for example G-d's name, is only an external vessel (בלי גבול בגבול היינו עצמות) associated with the inner being (בלי גבול היינו מהות). But to others (סביבה all created, finite beings בעלי גבול), the inner being מהותו יתברך is unknowable in a true sense. Our perception is limited by the fact that we are separate and finite beings (בעלי גבול). We can only know the name at present. And this is in keeping with what is expressed in Shemot 33:20 which says: “And (G-d) said it is not possible to see My face (inner essence מהות) because human beings cannot see Me and be alive.”
ויאמר לא תוכל לראת את־פני כי לא־יראני האדם וחי
But like we say in connection with every blessing that is recited after mentioning G-d's name, ברוך הוא וברוך שמו “Blessed is He (His Being מהותו) and blessed is His name." And like in Zechariah 14:9, this is what is meant when saying that G-d and His name are one.
יהוה אחד ושמו אחד
That the concept of Evil, meaning the perception and appearance of separateness from G-d, is a consequence of and arises from G-d's name. And His name exists in order that G-d be known, דעת כמו אדם ידע חוה so that there is a way to connect with Him (meaning His being מהותו) and ultimately be one with G-d.
This is what is expressed in the words of the Adon Olam prayer:
אֲדוֹן עוֹלָם אֲשֶׁר מָלַךְ, בְּטֶֽרֶם כָּל יְצִיר נִבְרָא: לְעֵת נַעֲשָׂה בְחֶפְצוֹ כֹּל, אֲזַי מֶֽלֶךְ שְׁמוֹ נִקְרָא: וְאַחֲרֵי כִּכְלוֹת הַכֹּל, לְבַדּוֹ יִמְלֹךְ נוֹרָא: וְהוּא הָיָה וְהוּא הֹוֶה, וְהוּא יִהְיֶה בְּתִפְאָרָה: וְהוּא אֶחָד וְאֵין שֵׁנִי, לְהַמְשִׁיל לוֹ לְהַחְבִּֽירָה: בְּלִי רֵאשִׁית בְּלִי תַכְלִית, וְלוֹ הָעֹז וְהַמִּשְׂרָה:
And this is what is meant from the expression found in Midrash Tanchuma, parshat Nasso, Siman 16 that said G-d desired for Himself a temporary dwelling place in the lower realms...
בְּשָׁעָה שֶׁבָּרָא הַקָּדוֹשׁ בָּרוּךְ הוּא אֶת הָעוֹלָם, נִתְאַוָּה שֶׁיְּהֵא לוֹ דִּירָה בַּתַּחְתּוֹנִים כְּמוֹ שֶׁיֵּשׁ בָּעֶלְיוֹנִים.
Temporary dwelling place, in Hebrew דירה, is the creation of Time (Temporality), expressed in Torah literature as the concept of שנה and Space, in Hebrew the concept of מקום or עולם, which means the two concepts together (time and space) meaning the entire universe.
And the concept of impulse or desire, that G-d desired נִתְאַוָּה, means the Lower Domain is rooted in and coming from a source that transcends intellect שכל and purpose תכלית.
That the earth הארץ is associated with the Lower Realms, also called Depths תהום, like it states in the second sentence of the Torah.
והארץ היתה תהו ובהו וחשך על־פני תהום
That תהו ובהו is the joining together of תהו (meaning desolation or void like in the מקום פנוי) and בהו (chaos). בהו being 2 times 5 (ב פעמים ה) and 5 times 6 (ה פעמים ו) resulting in 40 (מ) which stands for King and Kingship מלך מלכות, and when together with תהו becomes Depths תהום. And if looking at the inner letters פנימיות of תהום, meaning ה״ו, which are also the middle, inner two letters from G-d's name יהוה, their numerical value when including the number of letters (11 plus 2) is 13, which is the Small Sum value מספר קטן of Darkness חשך 13. And 13 is also the numerical value of One אחד. And the letter Tav ת represents both life תחיה and its opposite תמות like is found in tractate Shabbat 55a.
אמר רב: ״תיו״ — תחיה, ״תיו״ — תמות.
This also pertains to the seal of the Holy One, blessed be He mentioned by Rabbi Chanina in tractate Sanhedrin 64a which says:
נפל להו פיתקא מרקיעא דהוה כתיב בה אמת האמר רבי חנינא שמע מינה חותמו של הקדוש ברוך הוא אמת
That Truth אמת is the Notarikon of אחד-מלכות-תחיה (One, Kingship and Life) meaning מהותו which shares the letters of תהום and additionally אמת is the Notarikon of אורייתא-משנה-תלמוד, the three parts of the Torah, which is His name עצמותו.
And this idea follows the language of the Tzidkat HaTzaddik quoted above concerning the Root of all Roots is הגיאות Pride. That one of the meanings of Pride is: delight or elation arising from some act, possession, or relationship as in Parental Pride.
And this is G-d’s intention with Creation and also the source of Evil, meaning in accordance with G-d’s will which is good only, like is mentioned above, that the concept of Evil רע in and of itself, is pointless and without purpose (it transcends intellect) and was part of the very beginning, at the point of differentiation in the Creation in order to manifest G-d's Kingship, like is found in the Sefer Yetzirah mentioned above.
I can understand and relate to your question from a Torah, philosophical, kabbalistic, and practical point of view as to why evil exists altogether .
I think western religions really have kind’ve screwed up the way we think about things in black and white and the term good and evil seems to be differentiated amongst each specific dogma in religion.
Simply put; every tribe, culture, religion, etc. has their own unique view of which ACTS are considered either good or evil from a moral, and ethical standpoint without understanding what the essence of it really means.
From a mystical standpoint, not only in Kabbalah but as a universal concept, there is some form of animalistic Passion and need for survival (Nefesh HaBahamis) that each individual human struggles to over come.
I would argue that the term “Evil“ (which has a negative connotation) is connected to the side of intuition, passion and desire. That this helps us to survive as individuals and is actually the opposite of death. It is actually about SELF-preservation, and SELFISH desires which blind people from seeing the consequences of their actions beyond the direct moment.
So then I consider, on the other hand, evil is connected to someone with sociopathic tendencies who is cold and emotionless, who doesn’t do things in the heat of passion, someone who makes their evil plans in a very calculating manner and detached from passion.
Even though they are seemingly opposite definitions, they are both connected to the sides of selfishness and separateness.
As we know from chassidus, in order to become a vessel capable of containing and ABSORBING AND INSULATING SPIRITUAL ENERGY, we need to be “Tov”. This allows us to NOT chase after our desires but instead to manifest positive, GOOD energy by attracting energy more passively through gratitude.
As Jews, Torah and mitzvahs allow us to RECEIVE AND ATTRACT goodness. This connects our desires to Hashem and brings achdus to humanity through COMPASSION, instead of just passion .
Through these intentions, we can also create a lens to see hashgacha pratit in the world and bring Mashiach.
As to WHY evil exists, perhaps the separateness allows individuals to have their own sense of perspective and opinions. This allows for people to come up with so many different answers to this question and see the value in other peoples opinions. And this is in order to bring peace, harmony and GOODNESS to the world.
It’s all about our INTENTIONS. This way we have a choice to overcome “evil” desires. Otherwise, what is the purpose of our existence in the physical world?
G-d did not create evil. He also does not emit evil, nor does He prevent it from happening (allowing it to exist) because G-d is not involved in human affairs. Rambam felt that evil is the result three things: when people harm themselves, when people harm others, and natural law, although good for the earth as when a hurricane cleans the atmosphere, it can harm or even kill individuals residing near the proximity. This is because the world was not only created for people. We are not the center of the universe. People often feel perplexed as to why bad things happen to good people. They feel that G-d should be like a loving father, but once these misconceptions are deleted, all problems with evil are removed. Also, a spider catching a fly is purely by chance.