What do the last two words of the verse, al panai, mean? N. Leibowitz (1976a, pp. 316-319) reviews three different ways the phrase has been understood.
One, (see Rashi and Fox 1995, p. 369) the phrase signifies G-d's presence. This would seem to limit the prohibition of believing in other gods to only being in G-d's presence (see Seforno?), and then this would be a partial recognition of the other gods. However, she notes "of course this is not so since He is omnipresent in both time and space." Yet, what then is the point of this phrase? Rashi quoting the Mechilta suggests that one might have thought that the prohibition was only applicable for the generation who left Egypt, but once the phrase al panai is added, which signifies G-d's presence, which is forever, then we know that the prohibition is binding on all generations. This is a difficult drasha since why would one have thought that the prohibition was only for the generation who left Egypt? And, if the point of the phrase is to teach that the prohibition is forever, it would have been simpler to say that directly.
A second approach to understanding the phrase al panai is that the phrase means "besides Me." (N. Leibowitz quotes this in the name of the Sefer Mitzvot Gadol, and this understanding appears in the translations by JPS, Sarna, 1991, p. 109 and Alter 2004, p. 428). With this understanding, it is also unclear what is the need for phrase since the verse could have just said "You are not to have other gods" without the phrase "Besides me."
A third approach is from Luzzatto that the phrase al panai refers to G-d being angry, and the phrase is a warning to the people that if they worship other gods then this will make G-d angry. With this understanding, the phrase al panai has a similar message to the end of 20:5.
My guess is that phrase "al panai" means that one is not to believe that other gods are equal to G-d. With this understanding, 20:3 does not prohibit the belief in other gods, as argued by the Rambam (above), but it prohibits one from believing that they are equal to G-d.
A second approach to understanding the phrase al panai is that the phrase means "besides Me." (N. Leibowitz quotes this in the name of the Sefer Mitzvot Gadol, and this understanding appears in the translations by JPS, Sarna, 1991, p. 109 and Alter 2004, p. 428). With this understanding, it is also unclear what is the need for phrase since the verse could have just said "You are not to have other gods" without the phrase "Besides me."
A third approach is from Luzzatto that the phrase al panai refers to G-d being angry, and the phrase is a warning to the people that if they worship other gods then this will make G-d angry. With this understanding, the phrase al panai has a similar message to the end of 20:5.
My guess is that phrase "al panai" means that one is not to believe that other gods are equal to G-d. With this understanding, 20:3 does not prohibit the belief in other gods, as argued by the Rambam (above), but it prohibits one from believing that they are equal to G-d.
The idea that other gods exist but that they are inferior to G-d occurs repeatedly in the Torah. For instance, Yitro states "now I know that G-d is greater than all the other gods" Shemot 18:11. In the song by the Sea, Shemot 15:11 records that Moshe said (and we recite the verse three times in our daily prayers) "who is like You amongst the gods? (Alter, 2004, p. 400 on Shemot 15:11 writes the Torah "had no difficulty in conceding the existence of other deities, though always stipulating their absolute inferiority to the God of Israel.") Avraham in his conversation with Malki-Tzedek, refers to G-d as the highest god (Bereshit 14:22, and this verse is included in the first blessing of the shemoneh esrei), which signals an acceptance of other gods but G-d is greater than them. The acceptance of the existence of other gods also appears in Shemot 12:12, Shemot 18:11, Shemot 22:19, Bemidbar 33:4, Devarim 3:24, Devarim 10:17 (also recited in the first blessing of the shemoneh esrei), Devarim 13:8, and Devarim 29:25. In Devarim 32:12 the words "there was no god with Him" implies that there are other gods since the verse could have simply said "there are no other gods." (Is the term Azazel in Vayikra 16:8,10 another example? Another example in our prayers is by ein ke-elohanu.)
Devarim 4:35 (also Devarim 4:39) records that there is no other besides G-d, which might be interpreted to mean that one cannot conceive of the existence of other gods. However, the explanation of the verse is that there are no other gods like G-d since even if one believes in other gods, one must recognize that they are inferior to G-d. (Even the Rambam, Basic principles of the Torah 1:4, explains the verse that it means that there is no being like G-d, see Twersky, 1972, p. 44.) With this understanding, these verses do not repudiate the existence of other gods, and they accord with all the verses quoted above and our understanding of Shemot 20:3.
Of course, just because the Torah allows one to believe that other gods exist does not mean that they exist, only that one is not prohibited from thinking this way. Also, even if one thinks they exist, still these other gods cannot be worshipped in any way, Shemot 20:4,5; 34:14, and it is forbidden to mention their names, Shemot 23:13, since this might lead one to worshipping them.
Why was the Torah written from the point of view that there are other gods who are inferior to G-d? A possible answer is from the Rambam's (Moreh 3:32) explanation for why the Torah includes sacrifices, that the people of the generation who received the Torah could not conceive of a religion without sacrifices. Similarly, G-d could not have commanded the people not to believe in other gods since in those days the existence of all gods was an accepted fact. This seems to be the idea behind G-d's statement to Moshe by the sin of the golden calf, that the people are stiff or hard necked, Shemot 32:9. Luzzatto (on 32:9) explains that G-d was saying to Moshe that the people were attached to making and worshipping idols, and that the people were too stubborn to change their practice. Instead, the goal was to minimize this belief by demanding that the people accept that G-d was superior to these other gods and by forbidding worship these other gods, a lesson that was not learnt by the sin of the golden calf.
I discussed this idea in my synagogue, and a friend Harold Davies suggested another or complimentary reason. He suggested that it is natural for some people (even in modern times) to believe in forces in the world, and hence G-d did not prohibit these beliefs as long as people do not worship these forces and think they are equal to G-d. This idea relates to the further question what does it mean other gods? G-d is the supreme force who controls the world, and exists for eternity, while these other gods are supernatural forces that people believe in such as demons, angels, and fairies (tooth?).
Possibly then the understanding to all these reference to other gods in the Torah just boils down to semantics. If one calls then gods, as the Torah does, then it is permissible to acknowledge their existence, provided that it is understood that they are inferior to G-d and not worshipped. On the other hand, if people want to label them as "forces," which immediately makes them inferior to G-d, then it is forbidden to acknowledge other gods since with this approach all gods are considered equal. Presumably those people who believe in angels and do not believe in the existence of other gods (following the Rambam's interpretation of the 20:3) follow this second approach.
In conclusion, in the Torah and in our prayers, we clearly refer to the existence of other gods. If this understanding bothers a person, then he/ she can label all the references to other gods as forces. However, the crucial point is that Shemot 20:3 does not prohibit people from believing in other gods or forces if they want to, just that they cannot think that they are equal to G-d, and Shemot 20:4,5 prohibits them from being worshipped.
Devarim 4:35 (also Devarim 4:39) records that there is no other besides G-d, which might be interpreted to mean that one cannot conceive of the existence of other gods. However, the explanation of the verse is that there are no other gods like G-d since even if one believes in other gods, one must recognize that they are inferior to G-d. (Even the Rambam, Basic principles of the Torah 1:4, explains the verse that it means that there is no being like G-d, see Twersky, 1972, p. 44.) With this understanding, these verses do not repudiate the existence of other gods, and they accord with all the verses quoted above and our understanding of Shemot 20:3.
Of course, just because the Torah allows one to believe that other gods exist does not mean that they exist, only that one is not prohibited from thinking this way. Also, even if one thinks they exist, still these other gods cannot be worshipped in any way, Shemot 20:4,5; 34:14, and it is forbidden to mention their names, Shemot 23:13, since this might lead one to worshipping them.
Why was the Torah written from the point of view that there are other gods who are inferior to G-d? A possible answer is from the Rambam's (Moreh 3:32) explanation for why the Torah includes sacrifices, that the people of the generation who received the Torah could not conceive of a religion without sacrifices. Similarly, G-d could not have commanded the people not to believe in other gods since in those days the existence of all gods was an accepted fact. This seems to be the idea behind G-d's statement to Moshe by the sin of the golden calf, that the people are stiff or hard necked, Shemot 32:9. Luzzatto (on 32:9) explains that G-d was saying to Moshe that the people were attached to making and worshipping idols, and that the people were too stubborn to change their practice. Instead, the goal was to minimize this belief by demanding that the people accept that G-d was superior to these other gods and by forbidding worship these other gods, a lesson that was not learnt by the sin of the golden calf.
I discussed this idea in my synagogue, and a friend Harold Davies suggested another or complimentary reason. He suggested that it is natural for some people (even in modern times) to believe in forces in the world, and hence G-d did not prohibit these beliefs as long as people do not worship these forces and think they are equal to G-d. This idea relates to the further question what does it mean other gods? G-d is the supreme force who controls the world, and exists for eternity, while these other gods are supernatural forces that people believe in such as demons, angels, and fairies (tooth?).
Possibly then the understanding to all these reference to other gods in the Torah just boils down to semantics. If one calls then gods, as the Torah does, then it is permissible to acknowledge their existence, provided that it is understood that they are inferior to G-d and not worshipped. On the other hand, if people want to label them as "forces," which immediately makes them inferior to G-d, then it is forbidden to acknowledge other gods since with this approach all gods are considered equal. Presumably those people who believe in angels and do not believe in the existence of other gods (following the Rambam's interpretation of the 20:3) follow this second approach.
In conclusion, in the Torah and in our prayers, we clearly refer to the existence of other gods. If this understanding bothers a person, then he/ she can label all the references to other gods as forces. However, the crucial point is that Shemot 20:3 does not prohibit people from believing in other gods or forces if they want to, just that they cannot think that they are equal to G-d, and Shemot 20:4,5 prohibits them from being worshipped.
Bibliography:
Abravanel (1437-1508), 1997, Commentary on Shemot. 1999, Commentary on Devarim, Jerusalem: Horev.
Alter, Robert, 2004, The five books of Moses: A translation and commentary, New York: W. W. Norton and Company.
Fox, Everett, 1995, The Five Books of Moses: A new translation, New York: Schocken Books.
Leibowitz, Nehama (1905-1997), 1976a, Studies in Shemot, translated by Aryeh Newman, Jerusalem: The World Zionist Organization.
Sarna, Nahum (1923-2005), 1991, The JPS Torah Commentary: Exodus, Philadelphia: The Jewish Publication Society.
Tigay, Jeffrey H., 1996, The JPS Torah Commentary: Deuteronomy, Philadelphia: The Jewish Publication Society.
Twersky, Isadore, 1972, A Maimonides reader, New York: Behrman House.
Fox, Everett, 1995, The Five Books of Moses: A new translation, New York: Schocken Books.
Leibowitz, Nehama (1905-1997), 1976a, Studies in Shemot, translated by Aryeh Newman, Jerusalem: The World Zionist Organization.
Sarna, Nahum (1923-2005), 1991, The JPS Torah Commentary: Exodus, Philadelphia: The Jewish Publication Society.
Tigay, Jeffrey H., 1996, The JPS Torah Commentary: Deuteronomy, Philadelphia: The Jewish Publication Society.
Twersky, Isadore, 1972, A Maimonides reader, New York: Behrman House.