Not "Heart" but "Feelings"
The word “heart” is correctly translated in the Greek New Testament, but we can understand what it means more clearly with a little exploration. The Greek word for "heart" is kardia (καρδία). It means both the physical organ, and the seat of “higher” emotions, both passion and anger. Jesus uses kardia in thirty-seven verses.. He references it more than any other component of human nature other than pneuma, which means “breath” or “spirit,” and is used to refer to other “spirits” aside from the spirit within us.
Kardia also means “motivation” and “purpose” since emotions give people the desire to act. Jesus saw it as the motivation for both the positive and negative, which he describes in Luke 6:45:
KJV: A good man out of the good treasure of his heart bringeth forth that which is good; and an evil man out of the evil treasure of his heart bringeth forth that which is evil: for of the abundance of the heart his mouth speaketh.:
Notice how the first sign of what is in a person’s heart is in his or her words. These words lead to actions, as described in Matthew 15:19:
For out of the heart proceed evil thoughts, murders, adulteries, fornications, thefts, false witness, blasphemies:
Note that in this verse, Jesus views the human heart as inherently flawed. It gives rise to "worthless thoughts." The heart is the cause of thought, not the thoughts themselves. The spirit experiences thoughts in the mind through the urgings of the heart.
Shared Feeling
One interesting aspect of kardia is that Jesus often uses the singular, "the heart," to describe the thoughts or feelings of a group. This is typical of Jesus, indicating that he sees "the heart" as a shared feeling for a group. For example, he uses the plural possessive "your" with the singular "heart" in John 14:1:
KJV: Let not your heart be troubled: ye believe in God, believe also in me.
NIV: Do not let your hearts be troubled. You believe in God; believe also in me.
The phrase is translated as "your heart/hearts" is ὑμῶν ἡ καρδία. This means literally, “yours, the heart. The “your” is plural, but “the heart” is singular. Jesus may have adopted this idea singular heart for a group of people from the Old Testament. We can see this when he quotes the Septuagint in Matthew 15:8,
KJV: This people draweth nigh unto me with their mouth, and honoureth me with their lips; but their heart is far from me.
Here Jesus is quoting Isaiah 29:13. His Greek is identical to the Greek of the OT Septuagint. The Greek words are καρδία αὐτῶν, literally, “a heart of theirs.” So its sense would have been more of a shared "emotion" or "source of emotion" than the physical heart. So "heart" describes a shared mindset.
Jesus more rarely uses the plural "hearts," but he does do it, for example in Matthew 18:35.
KJV: So likewise shall my heavenly Father do also unto you, if ye from your hearts forgive not every one his brother their trespasses.
Literal: So also, that Father of mine, the heavenly one, will do for you all. When you all don't want to let go, each person for that brother of his, out of those hearts of yours.
However, this is an exception, perhaps because it refers to a personal choice. However, notice how difficult it is to see this idea in translation because of the problems of translating plural Greek into English. Notice also how “those hearts” are seen as a prison out of which we must free others.
Jesus’s common use of the singular for the group's "heart" indicates that Jesus recognized our feelings as arising from something greater than ourselves. "Heart" is connected both to mind and spirit. "Mind" is formed by our living within a social order, which is shared. Our feelings are, at least in part, learned from our environment. Similarly, our feelings are at least in part, generated from, "spirit," divine influence, which is also shared. We are not responsible for them or the thoughts they generate. We are only responsible for our actions: the choices made by our individual spirits despite the urging of our social hearts.
I am reminded of Huck Finn's battle conscience between what society taught him about slavery being good and right versus the urgings of his own spirit.
Related Greek Concepts
Kardia is also related to an important Greek concept Jesus only uses once, stethos ( στῆθος), which means "breast." The same concept is more frequently called thumos, (θυμός), which literally means "chest." This is another Greek concept of "soul," "spirit," "feeling and thought," and especially "of strong feeling and passion." Physically, the breath of pneuma goes into the breast or chest and from there to the heart. So, the heart is where spirit and mind are united. Lower animals are driven more by the belly not by the higher feelings of the chest.
This Greek concept of thumos or "feelings of the chest" or, as we might say, feelings of the heart. These included the love of family and country, hatred of enemies, anger, passion for learning, etc. Jesus seems to use kardia to describe all of this, neither good nor bad in themselves.
Conclusions
When Jesus uses the word “heart,” the concept shares many ideas we find today in English. This is probably because many of our ideas today come to us through the Bible. However, Jesus also used the term more broadly, the refer to the shared mindsets of groups of people. In reading Jesus in translation, we cannot always tell if he is referring to a personal feeling or a group feeling. This is because we often cannot tell the singular from plural in English translation.