In my statement of beliefs, I describe the verb "believe" as "accepting or receiving something as being true." In a verse like John 3:16, we believe in Jesus by receiving His testimony as being true.
Opponents of Free Grace Theology often say that believing with the heart is different than believing with the mind. Bob Wilkin of the Grace Evangelical Society has written a short article showing that the New Testament makes no distinction between "head belief" and "heart belief." His article should be sufficient to show that "believing with the mind" is the same as "believing with the heart," but he did not deal with Acts 8:37 where Philip speaks of "believing with all the heart." Acts 8:37 is not found in the majority of New Testament Greek manuscripts, so Wilkin doesn't consider it to be Scripture. However, I do believe that Acts 8:37 is authentic, so in this article, I would like to comment on that verse.
In Acts 8:35-37, we read the following:
35 Then Philip opened his mouth, and began at the same scripture, and preached unto him Jesus. 36 And as they went on their way, they came unto a certain water: and the eunuch said, See, here is water; what doth hinder me to be baptized? 37 And Philip said, If thou believest with all thine heart, thou mayest. And he answered and said, I believe that Jesus Christ is the Son of God.
The first thing we must notice is that Philip speaks of "believing with all thine heart" in the context of receiving water baptism. The eunuch was saved when he believed the preaching of Philip in verse 35. As they continued to talk, their conversation probably shifted to water baptism. During that period in history, the nation of Israel was called to repentance in preparation of the kingdom. Water baptism signified both repentance and allegiance to Messiah. The eunuch, as a proselyte of the gate (i.e., one who was uncircumcised, but worshiped God through the mediatorial agency of Israel), naturally wanted to show his allegiance to Messiah, and therefore, suggested that he himself also be water baptized. What is interesting to note is that it seems that Philip didn't regard water baptism as a requirement for the eunuch. In other words, instead of commanding the eunuch to be baptized, it was the eunuch who was requesting it. I suppose that this was because water baptism was a Jewish ordinance and not a requirement for real Gentiles (as opposed to the Jewish diaspora who were living among the Gentiles). For example, even John the Baptist suggested this when he said, "And I knew him not: but that he should be made manifest to Israel, therefore am I come baptizing with water" (John 1:31). The Apostle Paul, also, suggested that water baptism was not part of his ministry (or at least not of primary importance) to the Gentiles when he said, "For Christ sent me not to baptize, but to preach the gospel: not with wisdom of words, lest the cross of Christ should be made of none effect" (1 Corinthians 1:17). In my opinion, water baptism of real Gentiles during the Acts period was kind of a temporary concession to the Jews in order to facilitate their acceptance of Jesus as their Messiah. During the Acts period, it was "to the Jew first, and then to the Gentile" (Acts 13:46, Romans 1:16, etc.) Other concessions to the Jews would have been the decrees prescribed by the Apostles in Jerusalem in Acts 15:28-29 and the circumcision of Timothy in Acts 16:3. When Jesus said to "Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost:" in Matthew 28:19, it is clear from passages such as Acts 11:19 that the Apostles had interpreted this command as pertaining to the Jews of the diaspora who were living among the nations.
Getting back to our main topic, when the eunuch suggested that he himself be water baptized, Philip replied by saying "If thou believest with all thine heart, thou mayest." As I mentioned above, the issue here is not salvation, but water baptism that signifies allegiance to Messiah in the context of Israel's national repentance. Philip was basically saying, "If you really believe, you can take this next step of discipleship and allegiance." Remember that Jesus commanded the Apostles to "teach all nation and baptize" (Matthew 28:19). The verb "teach" here actually means "to disciple." Making a public confession of faith through water baptism and entering discipleship is not a small step to take. Jesus laid out the requirements of discipleship in Luke 14:26-33. In those verses, He explicitly says that a person contemplating discipleship must "count the costs" (Luke 14:28). Sadly, there are some people who rush to become disciples even before they have believed (John 6:60-64), and Philip wants to prevent this type of situation from happening. Therefore, he wants to know, "Do you really believe?"
When Philip says, "If you believe with all your heart, you may," the eunuch responded by saying "I believe that Jesus Christ is the Son of God." Of course, this reminds us of John 20:31 where we read, "But these are written, that ye might believe that Jesus is the Christ, the Son of God; and that believing ye might have life through his name." It seems that during their conversion, Philip taught the eunuch how to receive eternal life through belief in Jesus Christ as the Son of God. So, when Philip ask "if he believes with all his heart," the eunuch is basically saying, "Do you remember what you just taught me? Yes, I believe it! I have eternal life! I believe that Jesus Christ is the Son of God!" And by saying this, Philip was assured that he really did believe and was not just trying to get baptized and enter discipleship in order to earn salvation through his own works.
In my opinion, "believing with all the heart" and "believing" are one and the same. Philip's question was a good one because I suspect that there are many people who say they believe John 3:16, but really don't. I was one of these people. Please read my testimony to see how God convicted me over my disbelief of John 3:16. In addition to my testimony, I would also like to mention a story about an encounter I had with a youth pastor when I was in college. At that time, I had just started reading the Bible. I said I was a Christian and started attending our campus church, including morning services. One day, the youth pastor sat down with me and said, "Do you believe that if you died right now, you would go to heaven as a result of Jesus' free gift for you?" I immediately answered "Yes!", but in my heart, I was full of doubts. I simply said, "Yes" because that was what I knew I was supposed to say. If the pastor had said to me at that time, "Do you believe with all your heart?", I suppose that I might have looked like a deer caught in the headlights. The problem was that I didn't really believe Jesus' promise. John 3:16 just seemed too easy. I thought, "It can't be this simple." But several years later, I finally believed it and knew with certainty that "if I died right now, I would go to heaven because of what Jesus did for me." "Believing with all the heart" means having assurance that what Jesus said is true. And assurance really is the essence of saving faith. When Jesus said, "he that believes in me has eternal life" (John 6:47), if you really believe this, then you can know for sure that you have eternal life.