In Psalms 51:12 it reads
Fashion a pure heart for me, O God; create in me a steadfast spirit.
Which heart is refering to? Is refering to the organ that pumps blood?
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It is not heart in the physical sense, but rather the desires etc. we have that lead us to sin which is rooted in the heart. So, we are asking that our heart should be of purer intent in order that we do not slip up. Or to put it simply, a heart that does not sin again (see the Metzudos Dovid)
בעבור שהזכיר כי חולל בעון. והטעם: התאוה הנטועה הביאתהו לחטא, יתפלל אל השם שיעזרהו על יצרו, שלא יחטא פעם אחרת חטא כזה, וזה לב טהור
Since it mentions that he has committed a sin. And the reason: The desire has taken root and brought him to sin, so he will pray to G-d to help him with his inclination so that he does not sin again with a sin like this - and this a "pure heart".
Similarly, see the Malbim who adds that sin stems from a desire which is tangible/has substance (חומר), And the iniquity comes by the perversion of the mind or the spirit which brings up images to the heart which leads a person to behave badly. Therefore, one who does not sin is regarded as having a "לב טהור" - "pure heart", and one who has not committed an iniquity has a "רוח נכון" - "a correct spirit". Thus, the Malbim says we ask for a "new heart" that will no longer desire evil and likewise a spirit that is not distorted by his past.
לב. כי כונתי טובה מעתה ואתה עזרני בה, וסמוך לבי ורוחי שיהיה טהור שלא ימשול בי יצר הרע עוד וזהו הבריאה והחידוש
Heart - Because my intention is good from now on and you helped me with it, and trust my heart and my spirit to be pure so that no evil desire will rule me anymore and this is the creation and renewal
Of course, ein mikra yotze midei peshuto, so it could certainly be taken to mean flesh and blood heart. If one has any problems with their heart, I wouldn't be surprised if they were recommended to say this perek by their Rav.
Generally speaking, Lev is referring to the seat of emotions in the neshama. Thought, one of the levushim of the neshama (see Tanya Ch. 4), also takes place in the heart, so it can refer to thoughts as well (see Smak, Ch. 11 for eg).
The Zohar quotes this verse and explains that David is asking for this so he should be able to study Torah.
I hope that helps. Either way, there are many many ways to expound beautiful pasuk.
I heard in a shiur, why are there "gates" of tefilla? Because we have walls around our heart. Hardly a person is able to reach adulthood without losing some of their innocence, their sensitivity, and their trust, and in order to protect themselves, they put up walls around their heart. This is why adults are more numb than children, their emotions are narrower, and less intense. Prayer has the power to open gates in the heart and allow them to feel again.
This is important, because rachmana liba bai (Menachot 13:10). Hashem wants your heart. It is important for a close, intimate relationship.
The yeitzer hara and the yeitzer tov also dwell in the heart (see Tanya Chapter 13, 15 etc), and vie for control. Our mind is sometimes referred to as "mothers" and our emotions as "children" (see Tanya Chapter 3, or Sefer Yetzira for example), as all emotions are born in the mind. If we have learned Torah and absorbed it, then our mind will produce correct, pure emotions for goodness rather than evil (this is another connection to gates, as Binah, Understanding, has 50 gates, corresponding to the 49 kabbalistic emotions [see Rosh Hashana 21b, Maamar Chamishim Shaarei Binah 5653 etc], and each of these gates can be either pure, or impure, or even shut chas veshalom).
Hashem should purify all our hearts to serve Him, and eachother, with a good love, awe and compassion, with determination and gratitude, and with a sincere, powerful intimate connection.