Whoa, a six month old that does not sleep during the day. He needs to nap, he's too little not to.
First, get him in a routine, if you don't have him on one. He should be eating breakfast, snack, nap (around 9:30)this nap shouldn't be for a long period of time, just a power nap.
*try putting him in his swing in a low light area where you can check on him without him seeing you. Or put him in his playpen in a low light area, put a Baby Einstein movie on a relaxing one, not a stimulating one.
When he wakes it should be mealtime (lunch), playtime, nap (this nap should be the longest), when he wakes snacktime, playtime, Supper, bath between 6 and 7. Baths tend to either relax them or wake them up. If you do baths between these times it gives them time to wind down. Once out of tub, rub down with baby lotion to relax them and then it's playtime after your finish dressing, etc. Before bed give one more bottle with 4-6 ounces milk and this will tide him over for the night. If he wakes in the middle of the night, go and retuck him, give him his cuddly bear, etc., don't pick him up unless he's really throwing a fit, little talking if any and softly. Then go back to your bedroom. No matter how many times he wakes do the same thing. He will realize that your not going to pick him up eventually and drift off to sleep. He may be one that needs noise to sleep, try a noise maker, or he may need quiet, or he may need extra cushion in his crib, just make sure it's cushioned under the mattress to prevent suffocation. STICK TO YOUR ROUTINE!!! Now is the time your setting routine, because when he gets older he will need it.
It may take a little while to get him in this routine but after a time he will get used to it.
Good luck.
A.