The short answer is that God is spirit (John 4:24). Since God is spirit, He has no body, and therefore cannot be seen.
Moses “saw” God
However, in the Old Testament, at various points a few people did see God. In Exodus 33:18, Moses says to God “Show me your glory.” The LORD responded by saying, “I will cause all my goodness to pass in front of you, and I will proclaim my name, the LORD, in your presence. I will have mercy on whom I will have mercy, and I will have compassion on whom I will have compassion. But, you cannot see my face, for no-one may see me and live” (Exodus 33:19-20). What happened next is that the LORD placed Moses in a cleft in the rock and covered him with His hand, so that Moses could only see the LORD’s back (Exodus 33:21-23). Exodus 34:5-7 recounts the actual event:
Then the LORD came down in the cloud and stood there with him and proclaimed his name, the LORD. And he passed in front of Moses, proclaiming, “The LORD, the LORD, the compassionate and gracious God, slow to anger, abounding in love and faithfulness, maintaining love to thousands, and forgiving wickedness, rebellion and sin. Yet he does not leave the guilty unpunished; he punishes the children and their children for the sin of the fathers to the third and fourth generation.”
So Moses “saw” the LORD – or rather he saw the cloud representing God in all his holiness, splendor and majesty, and he heard the LORD proclaim his character to Moses.
Isaiah also “saw” God
In Isaiah 6, Isaiah had a similar experience:
In the year that King Uzziah died, I saw the Lord seated on a throne, high and exalted, and the train of his robe filled the temple. Above him were seraphs, each with six wings: With two wings they covered their faces, with two they covered their feet, and with two they were flying. And they were calling to one another: “Holy, holy, holy is the LORD Almighty; the whole earth is full of his glory.” At the sound of their voices the doorposts and thresholds shook and the temple was filled with smoke. “Woe to me!” I cried. “I am ruined! For I am a man of unclean lips, and I live among a people of unclean lips, and my eyes have seen the King, the LORD Almighty.” Then one of the seraphs flew to me with a live coal in his hand, which he had taken with tongs from the altar. With it he touched my mouth and said, “See, this has touched your lips; your guilt is taken away and your sin atoned for.”
Isaiah’s experience is very similar to Moses’. Isaiah saw the LORD (Isaiah 6:1), and yet all he really saw was “the train of his robe” which filled the temple.
God is holy
In both cases, neither Moses nor Isaiah really saw the LORD in his fullness, because “no-one may see [the LORD] and live” (Exodus 33:20). This is why Isaiah was full of fear: “Woe to me! I am ruined! For I am a man of unclean lips … and my eyes have seen the LORD” (Isaiah 6: 5).
The LORD is holy, righteous, just (Leviticus 19:2; Psalm 11:7; Revelation 15:3), and nothing impure can come before him or enter his presence (Leviticus 10:1-3; 1 Samuel 6:19-20; 2 Samuel 6:6-7). Therefore, if anyone does see the LORD, they will be put to death, because God is holy and we are not.
However, if you had lived in Israel about 2000 years ago, you could have seen God! The New Testament teaches that (John 1:1-2). God the Son took on human form and was born and dwelt on earth (John 1:14). As John writes, “That which was from the beginning, which we have heard, which we have seen with our eyes, which we have looked at and our hands have touched – this we proclaim concerning the Word of life” (1 John 1:1). John, the apostles, and many others, heard Jesus speak to them, saw him, and even touched him! However, when Jesus dwelt on earth, his glory was hidden, except at the transfiguration, (Matthew 17:1-13; Mark 9:2-13).
Although people still couldn’t see God the Father, we now “know” Him because Jesus has revealed Him to us. “And we know that the Son of God has come and has given us understanding, so that we may know him who is true; and we are in him who is true, in his Son Jesus Christ. He is the true God and eternal life” (1 John 5:20). “No one has ever seen God; the only God, who is at the Father’s side, he has made him known” (John 1:18).
We will all see God
Ultimately, when God ends this present world and creates the new heavens and earth (Revelation 21:1), all his people will see God face to face, “they will see his face, and his name will be on their foreheads” (Revelation 22:4). At that point in time, we will be able to look on God because we will be holy and pure. Those who are not will never enter God’s new creation (Revelation 21:27).
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