Maybe Pineapple Weed? It looks to be the right size, has the right kind of leaves, the flowers I’m not yet sure on. (They aren’t shown very well here.) Did I stumble on it?
Pineapple weed doesn’t have any petals at all. The flowers are greenish yellow.
Ok, the second link seemed to show white petals there.
Bleh, meant to also say, I’ve seen chamomile when the petals have fallen off, so I wondered if the one picture was just after the petals fell. 
No, those are just partly opened flowers.
Red flower: yep,Monarda didyma Smell the leaves, they have an oregano-ish fragrance, good ID for the plant before bloom.
White Flower: Pretty sure this is Feverfew,Tanacetum parthenium Here’s a link: http://davesgarden.com/pf/showimage/49946/ Yours has a showier flower, but there are some showy cultivars. There are some other photos on that page as well. The leaf will smell kind of bitter/musty, like a chrysanthemum.
I can buy that, especially if it’s a cultivar. Here’s a “double-flowered” cultivated variety.
Here’s more detail on the leaf: http://www.insectphotos.net/flowers/families/asteraceae/feverfewl.jpg
That leaf was reminding me of Tansy (another Tanacetum), but Tansy is yellow. I’m glad you came up with feverfew.
Colibri, I almost referred to 'Flore Pleno", but mack’s flowers have much bigger discs. Not to say that there’s not a cultivar with bigger discs. Lots O feverfews out there. Leaf does match, though. Plus, in garden sleuthing: it is a showy Monarda, so whoever gardened there previously might have also planted a showy Tanacetum in an herb garden.
I have a smidgen of doubt that it could be some early-blooming chrysanthemum, (and feverfews have been labelled chrysanthemum). My own test would be to taste a leaf. Feverfew has a very distinct, complex bitter taste, different than a chrysanthemum.
Responsible Disclaimer: don’t put plants in your gullet without proper ID, or knowledge thereabouts.
Interesting. The property is in a little hollow in the Catskill mountains and as far as I know it was only used long ago for grazing livestock. We’re the first ones to really build on it. It doesn’t seem like the kind of place where someone would go to all that trouble, but who knows!
There are some gardens around. Maybe a deer munched on some of this stuff and pooped the seeds out on my place.
OK, as long as you’re sure it’s not poison hemlock. 
Thanks for all the help, folks. I brought up a guide online and realized I didn’t even know where to start.
mack, see, that matters with this ID. Is it an old farmhouse, likely to have an herb gardener in past times? Both Monarda and Tanacetum are very aromatic, planted in herb gardens, but aromatics are most often avoided by deer(Smart plants!), so unlikely that seed is spread that way.
Too, Monarda is native, but yours is showy enough for me to think it was intentionally planted. Feverfew is not native, so was most probably planted. The two together indicates a past gardener with an interest in herbs.
The way it’s laid out is that the old homesite (just a flat spot of land now) is above our field, across the road. There was a barn at the very edge of the road on our side, then there’s an embankment that drops down from the road maybe 30 feet or so to the field where our house (three years old) is. The field is essentially bottomland. There may have been a chicken coop there. There are also some apple trees bordering the field.
The Monarda is in a small meadow off the the main field by a stone wall and the feverfew is at least a couple hundred feet away, about 1/3 of the way up the embankment, so it’s kind of random. Neither is really anywhere near the old barn site.
Out of curosity, would a cat eat this stuff? Because this spring a feral cat hung out in gaps in the stone wall by where the monarda is and we also spotted it on the embankment near where the Feverfew is.
I’m starting to realize how stupid this dung theory is. Carry on. Beddy-bye time.
Hmmm, don’t know if cats eat Monarda or Feverfew (using this to distinguish from tansy nothin’ bothers that) They do love Nepeta, catnip, also about as strong aromatically. Really doubt that would be a factor at seeding in, though.