Your first question upon seeing the title to this might be "Why does this matter to me?"
Well, if you're a workaholic, rarely see your kids, and pretty much let your wife handle decisions concerning them 99% of the time...it probably won't matter that much to you. For those more involved fathers though, this has a significant impact during the first year of your child's life, and chances are if you are involved with your children, you will also have a fair amount of influence in your wife's decision of whether or not to breastfeed, especially if you appear to be informed about the topic.
Now first off - by no means is this a clear-cut choice. Despite what the Breastfeeding Nazi's, aka "Le Leche League" will tell you, neither method is wholly superior to the other. So I'll basically go over the upsides and downsides of Formula VS Breastfeeding. Since any upside to breastfeeding is a downside to formula and vice versa, I will just go over it from breastfeeding being the "default", and you can put 2 and 2 together and figure out which is better for you and your family.
As a side note - if your wife happens to know any breastfeeding Nazi's - be warned, they will be VERY opinionated as to how YOUR child is fed. Resolve this however you feel is most prudent - personally, I prefer the "Extremely rude tell-them-to-fuck-off" approach. This may not work so well depending on who the person happens to be to your wife though.
Breastfeeding -
Positives -
1 - From your perspective, breastfeeding takes a load of you, a lot of the work is shifted to the mother for a breastfed child. This may or may not be a positive to you though - perhaps you prefer to be more involved. For myself, since I was taking care of the other children while my wife cared for the infant, this worked out fairly well.
2 - A Breastfed child is generally easier to comfort. You stick a boob in their mouth, and they'll calm down around 90% of the time. And the other 10% they don't, there is usually a problem you can fix, such as a diaper change. If baby bumps herself, put them on the boob, if baby is getting overwhelmed, put her on the boob...you get the idea. It is, for most children, somewhat of a cure-all.
3 - Skin issues. If you have any sort of skin issues or dryness in yours or your wife's family...this can be the most compelling reason by far to breastfeed. Look up "Eczema" if you want to see what I mean. Breastfed children have a much lower rate of Eczema, which, if your child ends up with it, can be a MAJOR pain in the ass. By that I mean, applying a type of skin cream EVERY day, sometimes twice a day, for years. And that only staves it off - your child will still suffer from skin irritation constantly.
4 - Cost. I think this one is self-explanatory. Formula isn't cheap.
5 - Unknown? There are many other possible long-term benefits of breastfeeding, however these are harder to quantify, as it us currently not 100% known how much breastfeeding impacts them. For example, there used to be a study that breastfed kids were smarter children overall, but it was found out to be untrue - breastfed children simply had smarter parents, so the benefit was actually genetic(although I suppose if smart people are doing it it may not be a bad idea, right?)
Negatives -
1 - If you want to develop an early bond with your baby as a father, breastfeeding may in fact hamper this. While you may love your baby(hopefully!), your baby may not be as fond of you, as you do not provide food for them. This may sound inconsequential, but to a baby, food and sleep are pretty much all they got going for them at that stage in life. My second daughter would have nothing at all to do with me for the first 6 months of her life(Although my 3rd there was no issue). The upside - if you're persistent, it won't matter, they'll come around eventually anyway.
2 - You know how I said a positive is that it lightens your own personal load? Well, this can be a negative as well, as it puts a large amount of extra burden on the mother. If your wife is easily stressed out, prone to depression, etc, it may be a safer bet to go with formula. Your wife is more prone to emotional problems in the months following giving birth, so take that into account.
3 - Breastfeeding may not always work. Some women simply don't produce enough milk, or the baby may have trouble latching, or any number of other things. This can also vary from pregnancy to pregnancy. Although formula can have it's own problems, there are many types of formula so it's usually possible to work around them.
4 - Travel, especially in early months, can be difficult. One major advantage of formula is you can whip it out anywhere, and even feed the child while moving or driving. For obvious reasons, this is not so easy with breastfeeding, especially if your wife is uncomfortable doing it in public.(there are items to help hide it by the way). As your child can go longer and longer periods between feedings, this becomes less of a problem.
5 - On the same note, while breastfeeding is the ultimate source of comfort, it's not as portable. Many breastfed children will also shun a pacifier, making it difficult to comfort them with anything but the boob.
6 - Sex. This can likely impact different women in different ways, but personally, I've found that breastfeeding tends to kill a woman's libido. This makes scientific sense, as a woman's body would want to avoid getting pregnant while breastfeeding, to avoid having to produce for 2 children at once. This is supported also by the fact your wife may not ovulate during breastfeeding for up to the first 6 months or so. Also - a woman who's breastfeeding tends to have sore nipples all of the time, so this kind of eliminates them as anything but a feeding device.
In conclusion - although it may seem there are more negatives than positives, not all have equal weight, so the overall is that it's simply a matter of what works best for you and your wife.
