Paul's list of the resurrection appearances in 1 Corinthians 15, likely derived from an early creedal tradition, is an important datum when analyzing Christian origins. While I think some apologists overstate the significance of this creed, it is nonetheless deserving of careful study as it gives us an account outside of and earlier than the Gospels from someone who claimed to have seen the risen Jesus.
A common objection to the appearances recounted in this creed is that the Greek word for "appeared", ὤφθη (passive form of ὁράω, "to see"), often denotes appearances in visions or other types of vague, spiritual manifestations. This, it is argued, casts doubt on whether the appearances of Jesus referenced in 1 Corinthians 15 are on the same footing as the robust, physical appearances recounted in the later Gospels. The purpose of this post is to lay to rest the notion, primarily through literary examples, that ὤφθη carries any relevant connotations beyond our English "appeared", and thus, it is hardly notable in analyzing the creed.
Here is some more information on ὤφθη, and the root verb ὁράω. A Greek–English Lexicon of the New Testament and Other Early Christian Literature, 3rd ed. (BDAG) gives the following definitions for ὁράω:
1) to perceive by the eye, catch sight of, notice
2) to see someone in the course of making a friendly call, visit
3) to experience a condition or event, experience, witness
4) to be mentally or spiritually perceptive, perceive
Under definition #1, the passive form of the verb is defined as "become visible, appear". The passive form is what we're concerned with in 1 Corinthians 15.
From the 10-volume Theological Dictionary of the New Testament (TDNT):
"The range of meaning is much the same in the case of ἰδεῖν ("to see") as in that of ὁράω. It denotes seeing as sense-perception, hence eye-witness."
Further:
"The important point about ὤφθη with the dative, however, is that the one who constitutes the subject is the one who acts, i.e., appears, shows himself, with no special emphasis on the resultant action of the person in the dative, namely, that he sees or perceives." (TDNT)
Instances in which ὁράω refers to normal seeing
For many of these examples I am indebted to NT Wright's book The Resurrection of the Son of God. I have divided the verses into sections which broadly delineate the different types of appearances involved. Most of the English translations are taken from the NASB. The Greek terms are taken from the Septuagint, specifically this online version.
Appearing before God
Deuteronomy 16.16
Three times a year all your males shall appear (ὀφθήσεται) before the Lord
your God at the place which He chooses: at the Feast of Unleavened
Bread, at the Feast of Weeks, and at the Feast of Booths; and they are
not to appear (ὀφθήσῃ) before the Lord empty-handed.
Deuteronomy 31.11
when all Israel comes to appear (ὀφθῆναι) before the Lord your God at the place which He will choose, you shall read this Law before all Israel so that they hear it.
Isaiah 1.12
When you come to appear (ὀφθῆναι) before Me,
Who requires of you this trampling of My courtyards?
Sirach 52.4
Thou shalt not appear (ὀφθῇς) empty before the Lord.
Appearing before other people
Genesis 46.29
And Joseph prepared his chariot and went up to Goshen to meet his father
Israel; as soon as he appeared (ὀφθεὶς) to him, Joseph threw himself on his neck
and wept on his neck a long time.
1 Kings 3.16
Then two women who were prostitutes came to the king and stood before him (ὤφθησαν).
1 Kings 18.1
Now it happened after many days that the word of the LORD came to Elijah
in the third year, saying, “Go, present yourself (ὄφθητι) to Ahab, and I will
provide rain on the face of the earth.”
Sirach 39.4
He shall serve among great men, and appear (ὀφθήσεται) before princes: he will
travel through strange countries; for he hath tried the good and the
evil among men.
1 Maccabees 4.6
But as soon as it was day, Judas showed himself (ὤφθη) in the plain with three
thousand men, who nevertheless had neither armour nor swords to their
minds.
1 Maccabees 4.19
As Judas was yet speaking these words, there appeared (ὤφθη) a part of them looking out of the mountain...
Acts 7.26
And on the following day he appeared (ὤφθη) to them as they were fighting each
other, and he tried to reconcile them to peace, by saying, ‘Men, you are
brothers, why are you injuring each other?’
Other
Judges 5.8
New gods were chosen;
Then war was in the gates.
Not a shield or a spear was seen (ὀφθῇ)
Among forty thousand in Israel.
Genesis 1.9
Then God said, “Let the waters below the heavens be gathered into one place, and let the dry land appear (ὤφθη)”; and it was so.
Leviticus 13.14
But whenever raw flesh appears (ὀφθῇ) on him, he shall be unclean.
Deuteronomy 16.4
For seven days no leaven shall be seen (ὀφθήσεταί) with you in your entire territory...
2 Chronicles 9.11
From the algum trees the king made steps to the house of the LORD and
for the king’s palace, and lyres and harps for the singers; and nothing
like them was seen (ὤφθησαν) before in the land of Judah.
Song of Solomon 2:12
The blossoms have already appeared (ὤφθη) in the land...
Exodus 10.28
Then Pharaoh said to him, “Get away from me! Be careful, do not see my
face again, for on the day you see (ὀφθῇς) my face, you shall die!”
The Significance of ὤφθη
The above examples (more could be given) should suffice in showing that when Paul says the risen Jesus "appeared" no more needs be read into it than that Jesus appeared in some way to his disciples. It is worth noting that even if ὤφθη has spiritual or visionary connotations, this is still applicable to appearances of the risen Jesus. Paul held a very high Christology (Philippians 2:5-11) so it's understandable that he would use such a word in reference to the appearances.
In conclusion, there are plenty of instances in which the passive form of ὁράω and even the specific passive form ὤφθη refers to a regular, physical appearance. Of what exactly Paul or the early church believed these appearances to consist is a matter that must be settled on other grounds.