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Bal Tashchit / Do Not Destroy
The Biblical Source for "Do Not Destroy"

(יט) כִּֽי־תָצ֣וּר אֶל־עִיר֩ יָמִ֨ים רַבִּ֜ים לְֽהִלָּחֵ֧ם עָלֶ֣יהָ לְתָפְשָׂ֗הּ לֹֽא־תַשְׁחִ֤ית אֶת־עֵצָהּ֙ לִנְדֹּ֤חַ עָלָיו֙ גַּרְזֶ֔ן כִּ֚י מִמֶּ֣נּוּ תֹאכֵ֔ל וְאֹת֖וֹ לֹ֣א תִכְרֹ֑ת כִּ֤י הָֽאָדָם֙ עֵ֣ץ הַשָּׂדֶ֔ה לָבֹ֥א מִפָּנֶ֖יךָ בַּמָּצֽוֹר׃ (כ) רַ֞ק עֵ֣ץ אֲשֶׁר־תֵּדַ֗ע כִּֽי־לֹא־עֵ֤ץ מַאֲכָל֙ ה֔וּא אֹת֥וֹ תַשְׁחִ֖ית וְכָרָ֑תָּ וּבָנִ֣יתָ מָצ֗וֹר עַל־הָעִיר֙ אֲשֶׁר־הִ֨וא עֹשָׂ֧ה עִמְּךָ֛ מִלְחָמָ֖ה עַ֥ד רִדְתָּֽהּ׃ (פ)

(19) When in your war against a city you have to besiege it a long time in order to capture it, you must not destroy its trees, wielding the ax against them. You may eat of them, but you must not cut them down. Are trees of the field human to withdraw before you into the besieged city? (20) Only trees that you know do not yield food may be destroyed; you may cut them down for constructing siegeworks against the city that is waging war on you, until it has been reduced.
Ruining Items and Loss of an Item Falls Under "Do not destroy"

אלא האי תנא הוא דתניא מקרעין על המת ולא מדרכי האמורי אמר רבי אלעזר שמעתי שהמקרע על המת יותר מדאי לוקה משום בל תשחית וכ"ש גופו ודלמא בגדים שאני דפסידא דלא הדר הוא כי הא דרבי יוחנן קרי למאני מכבדותא ורב חסדא כד הוה מסגי ביני היזמי והגא מדלי להו למאניה אמר זה מעלה ארוכה וזה אינו מעלה ארוכה

The Gemara suggests: Rather, it is the opinion of this tanna, as it is taught in a baraita: One may rend garments in anguish over one who died, and it is not considered of the ways of the Amorites, but a Jewish custom. Rabbi Elazar says: I heard that one who rends his garments excessively over one who died is flogged for having transgressed the prohibition of: Do not destroy (see Deuteronomy 20:19). The Gemara suggests: And all the more so it is the case that according to Rabbi Elazar one who injures his body in anguish transgresses this prohibition. The Gemara rejects this suggestion: But perhaps garments are different, in that tearing them is a loss that is irreversible, like that practice of Rabbi Yoḥanan, who would refer to his garments as: My honor, and like that practice of Rav Ḥisda, who, when he would walk among thorns and shrubs, would raise his clothing despite the fact that his skin would get scratched by the thorns. He said in explanation of his actions: This flesh will heal if scratched, but that garment will not heal if torn. Similarly, perhaps it is prohibited to rend one’s garments, but it is permitted to injure oneself.
Rabbinic Sources expanding "Do not destroy"

(י) וְלֹא הָאִילָנוֹת בִּלְבַד. אֶלָּא כָּל הַמְשַׁבֵּר כֵּלִים. וְקוֹרֵעַ בְּגָדִים. וְהוֹרֵס בִּנְיָן. וְסוֹתֵם מַעְיָן. וּמְאַבֵּד מַאֲכָלוֹת דֶּרֶךְ הַשְׁחָתָה. עוֹבֵר בְּלֹא תַשְׁחִית. וְאֵינוֹ לוֹקֶה אֶלָּא מַכַּת מַרְדּוּת מִדִּבְרֵיהֶם:

(10) And not only regarding trees, but even one who destructively breaks vessels/tools/utensils or rips up clothing or tears down a building or seals up a spring or wastes food violates the Negative Commandment of “Do not destroy”. However, they only receive lashes for disobedience, in accordance with the Rabbis.

(א) שהזהירנו מהשחית האילנות כשנצור על עיר כדי להצר לאנשיה ולהכאיב לבם. הוא אמרו יתעלה לא תשחית את עצה, וכן כל הפסד נכנס תחת לאו זה כגון מי שישרוף בגד לריק או ישבור כלי גם כן עובר משום לא תשחית ולוקה, ואמר אזהרתה מהכא כי ממנו תאכל ואותו לא תכרות.

That we are warned to avoid destroying trees when we besiege a city in order to distress its people and to cause them emotional pain. God said not to destroy its trees, and so too any any loss comes under this negative commandment, such as one who burns a garment for no reason or someone who breaks a vessel also violates "Do not destroy" and is lashed, and it gives a warning from here for "you shall eat from it - do not cut."...

Rabbi Samson Raphael Hirsch, 19th cent. Germany, Horeb, #56
"Do not destroy anything" is the first and most general call of God... If you should now raise your hand to play a childish game, to indulge in senseless rage, wishing to destroy that which you should only use, wishing to exterminate that which you should only exploit, if you should regard the beings beneath you as objects without rights, not perceiving God Who created them, and therefore desire that they feel the might of your presumptuous mood, instead of using them only as the means of wise human activity -- then God's call proclaims to you, "Do not destroy anything! Be a mentsh! Only if you use the things around you for wise human purposes, sanctified by the word of My teaching, only then are you a mentsh and have the right over them which I have given you as a human. However, if you destroy, if you ruin, at that moment you are not a human but an animal and have no right to the things around you.
Mishnaic Source about Upcycling

(ג) מִבְּלָאֵי מִכְנְסֵי כֹהֲנִים וּמֵהֶמְיָנֵיהֶן מֵהֶן הָיוּ מַפְקִיעִין, וּבָהֶן הָיוּ מַדְלִיקִין, וְלֹא הָיְתָה חָצֵר בִּירוּשָׁלַיִם שֶׁאֵינָהּ מְאִירָה מֵאוֹר בֵּית הַשּׁוֹאֵבָה:

(3) From the worn out pants of the priests and from their [worn out] belts they would tear [pieces], and they would [use them as wicks to] light with them. And there was not a courtyard in Jerusalem that was not illuminated by the light of the place of [water] drawing.

Issues in Jewish Ethics: Recycling in Jewish Tradition
by Rabbi Binyomin Adilman
For example, the lulav (willows and myrtles that are waved on Sukkot) are put away and later used for fueling the fire that burns the chametz (leavened products) on Erev Pesach. The etrog from Sukkot is poked full of cloves and used to make a fragrant pomander for the Havdalah service at the conclusion of Shabbat. The old worn tzitzit fringes, when replaced, are used as bookmarks in chumashim, etc. Olive oil from the Land of Israel is hung in the Sukkah as a representative of one of the seven species of the land. It is then saved to use for lighting the Chanukah lights. In addition, a small amount is squeezed out of the wicks, and is eaten six weeks later on Tu B'Shevat. A tallis is used for the chuppah at a Jewish wedding and then becomes the prayer shawl for the husband. Sometimes a portion of the wine over which the blessings are said during the marriage ceremony is saved and used for the cup of blessing at the brit milah ceremony of that same couple's son.
Biblical Source for Humanity's Role on Earth

(טו) וַיִּקַּ֛ח ה' אֱלֹקִ֖ים אֶת־הָֽאָדָ֑ם וַיַּנִּחֵ֣הוּ בְגַן־עֵ֔דֶן לְעָבְדָ֖הּ וּלְשָׁמְרָֽהּ׃

(15) The Eternal God took the human and placed it in the garden of Eden, to work it and to guard it.

Rabbinic Midrash Based on Genesis 2:15

(א) רְאֵה אֶת מַעֲשֵׂה הָאֱלֹקִים כִּי מִי יוּכַל לְתַקֵּן אֵת אֲשֶׁר עִוְּתוֹ, בְּשָׁעָה שֶׁבָּרָא הַקָּדוֹשׁ בָּרוּךְ הוּא אֶת אָדָם הָרִאשׁוֹן, נְטָלוֹ וְהֶחֱזִירוֹ עַל כָּל אִילָנֵי גַּן עֵדֶן, וְאָמַר לוֹ, רְאֵה מַעֲשַׂי כַּמָּה נָאִים וּמְשֻׁבָּחִין הֵן, וְכָל מַה שֶּׁבָּרָאתִי בִּשְׁבִילְךָ בָּרָאתִי, תֵּן דַּעְתְּךָ שֶׁלֹא תְקַלְקֵל וְתַחֲרִיב אֶת עוֹלָמִי, שֶׁאִם קִלְקַלְתָּ אֵין מִי שֶׁיְתַקֵּן אַחֲרֶיךָ...

(1) Look at God's work - for who can straighten what God has twisted? (Ecclesiastes 7:13). When the Blessed Holy One created the first human, God took the human and led him round all the trees of the Garden of Eden and said to him: “Look at My works, how beautiful and praiseworthy they are! And all that I have created, it was for you that I created it. Pay attention that you do not ruin and destroy My world: if you ruin it, there is no one to repair it after you.

אמר רב יהודה אמר שמואל כל הנהנה מן העולם הזה בלא ברכה כאילו נהנה מקדשי שמים שנאמר לה׳ הארץ ומלואה רבי לוי רמי כתיב לה׳ הארץ ומלואה וכתיב השמים שמים לה׳ והארץ נתן לבני אדם לא קשיא כאן קודם ברכה
Similarly, Rav Yehuda said that Shmuel said: One who derives benefit from this world without a blessing, it is as if he enjoyed objects consecrated to the heavens, as it is stated: “The earth and all it contains is the Lord’s, the world and all those who live in it” (Psalms 24:1). Rabbi Levi expressed this concept differently. Rabbi Levi raised a contradiction: It is written: “The earth and all it contains is the Lord’s,” and it is written elsewhere: “The heavens are the Lord’s and the earth He has given over to mankind” (Psalms 115:16). There is clearly a contradiction with regard to whom the earth belongs. He himself resolves the contradiction: This is not difficult. Here, the verse that says that the earth is the Lord’s refers to the situation before a blessing is recited,
Lawrence Troster, An Environmental Al-Het, from: Laurel Kearns & Catherine Keller, editors, EcoSpirit: Religions and Philosophies for the Earth, p. 524-525:
Lord, our Creator, we awaken each morning to the dawn chorus of Creation.
Our ears hear the birds of the sky singing to the world that they are still alive,
Our eyes see the flowers of the earth opening to the light of the sun.
We smell the scents of the fresh morning air.
How many are the things You have made O Lord, the universe is full of Your creations! And yet we ignore these sounds, sights and smells.
Instead of the birds’ song we hear only the sound of cars and machinery.
Instead of the sight of green, brown and gold we see only the gray of concrete.
Instead of the fragrance of flowers we smell only the stink of pollution.
We experience only the fruits of our own creations.
We know only of our own works which too often have wasted Your creation and silenced many of the voices of Your choir.
We think we understand the world when only a fool thinks they can fathom the depths of Your designs.
May You give us the strength and the wisdom to see, smell and listen to Your creation and be moved to protect and cherish the blessings that You have given us.
May we no longer be moved by greed and destruction to waste Your world for if we destroy it there will be no other.
We now know that the destruction of Your Creation is a sin.
And so for the sin that we have sinned against You by despoiling Your Creation, forgiving God, forgive us, pardon us, grant us atonement.