Rosh Hashannah 17b–
The Alshich in Shlach 14:17-20 says that the question has been asked that this Chazal contradicts our experience; plenty of times we say it over and over, and it doesn't do a thing! (An interesting question! Most teachers would react to a question like this with a frask in punim--"Who are you going to trust, you sheigetz, Chazal or your lying eyes???? A more confident didact would respond with the usual "Sometimes ‘No’ is the best answer", or "They can’t overcome a gzeira that was finalized with a shevu’ah", or that "The effect of the tefilla is hidden or pending".)
The Alshich brings an answer from the Livnas Sapir: The recital of the Thirteen Middos is only effective when the person who says them fulfills the Gemara in Shabbos 121 "mah hu rachum v’channun...," just as He is merciful, you too be merciful, just as He is long tempered, etc., that the person emulates the traits of God that are enumerated in the Thirteen Middos.
Another way of putting the Alshich’s (Livnas Sapir) teretz is "It is not enough to say the yud gimmel middos. You have to be the yud gimmel middos."
Imagine a father who is an achzar with his children, who says with great kavana "Oy, Avinu, Ha'av Harachaman, racheim aleinu!" He's a father, and he has no rachmanus on his children! Or, someone who is mean and heartless, who is davenning "ki eil melech chanun verachum atta" with real kavana and dveikus. How do you think the Ribbono Shel Olam should respond to such people? What good are such tefillos? If, on the other hand, you think, as you daven, that this is a middah that one should emulate and express in one’s life, and you determine to do so, then the tefilla would certainly mean a great deal more. The great chiddush of the Alshich is that this is true even regarding the Bris of the 13 Middos.
Even farther: the Tzror HaMor in Ki Sisa (34:9) says
Tefilla without incorporating into yourself what you hope to elicit from Hashem is not Tefilla at all. Tefilla is not an arm's length entreaty that, if heard and fulfilled, enables you to passively observe as your prayers are answered. Tefilla is part of a process that includes dveikus or inspired zimra and tehilla or bitachon, that enables you to become a fitting vessel for the achievement of Hashem's will. The answer to successful Tefilla is the opportunity to participate in the result. By becoming God-like, you can become a fitting conduit that allows the will of God to flow, through you, to this world. Energy can't flow through 'nothing.' There's got to be a medium that is hospitable to the energy to allow it to move. The medium is you. The least you need to do is to be a conductor, not an insulator.
We find the same concept in Nedarim 40a--
This pshat is supported by the Gemara in Rosh Hashanna 17b. There, the Gemara says that Hashem was mis'atef with a tallis like a shli'ach tzibur, and said the 13 Middos, and told Moshe "Im ya'asu banai lefanai kaseder hazeh Ani mochel lahem." The word "Ya'asu" implies action, even though the context is speech, tefillah by a shli'ach tzibur. But according to the above, the two concepts are congruous: the bris of the 13 Middos involves tefilla, speech, which reflects or generates action, the act of emulating Hashem.
A Commenter pointed out that the Alshich learned in Reb Yosef Karo's yeshiva, and he had a fellow talmid named Reb Moshe Cordevero, who wrote the Tomer Devora. The entire theme of the Tomer Devora is a discussion of the obligation, and how, to emulate Hashem's 13 Middos. He ends the first chapter by saying "Just as a man behaves below, so he will merit to open himself "Middah Elyonah Mil'e'maalah." Exactly as he behaves, "kach mashpi'ach mi'le'maalah," and he causes that Middah to shine in the world." With that in mind, it seems evident that this theme, as quoted by the Alshich from the Livnas Hasapir, was, among RYK's talmidim, an important and very carefully considered hashkafas hachaim.
This approach reminds me of the Nefesh Hachaim’s idea of "Hashem Tzilchah, " (see Nefesh Hachaim 1:7 DH V’zeh she’omar Dovid Hamelech Hashem Tzilchoh and the next paragraph, and 1:9 DH Omnom ho’inyon, and the Hagoho there who says the Anaf Yosef’s vort about Tapu’ach which I bring here, which R Chaim Volozhener also says klor in his pirush on Shir Hashirim 2:3) and what I wrote in Devarim Eikev 10:17:
An example of this: the Gemora in Shabbos 88a says, "Amar R’ Chamma bar Chanina, ‘ketapuach be’atzei hayaar...’, lama nimsh’lu Yisrael l’Tapuach?" etc. Rabbeinu Tam asks there, that in that passuk, it is not Klal Yisroel being compared to a Tapuach, but rather Hashem? The Anaf Yosef in the Ein Yakov brings down the Nefesh Hachaim that answers that if Klal Yisroel perceived and compared Hashem to ‘tapuach’, this is definitely because Klal Yisroel are comparable and the behavior is similar to ‘tapuach’, because "kederech she’anachnu misro’im lefonov yisborach, kach hu yisborach shemo bo lei’ra’os el ha’olamos al zeh ha’hadraga vehashiur mamash." From this vort you see that not only does Hashem appear in a fashion that reflects our behavior, but that what we call His semblance is just one of the ways that He communicates with us through His behavior.
הודעני נא את דרכיך ואדעך וכו' ,
והתבונן במה שנכלל בלשון זה מן המופלאות, אמרו 'הודעני נא את דרכיך ואדעך', מלמד על היותו יתעלה נודע בתאריו, שאם ידע הדרכים ידעהו. ואמרו 'למען אמצא חן בעיניך', מלמד כי מי שידע את ה', הוא אשר ימצא חן בעיניו. לא מי שצם והתפלל בלבד , אלא כל מי שידעו הוא הרצוי המקורב, ומי שסכל ידיעתו, הוא הזעום המרוחק. ולפי ערך הידיעה והסכלות יהיה הרצון והזעם והקרבה והריחוק.