Mokka mit Schlag » 2026 » January

I can’t believe I got this far in mathematics without noticing that not only can the parallel postulate be proved, but that it’s been proved for hundreds of years. For the last hundred or so, it’s been completely provable, along with the rest of Euclidean geometry, in Zermelo-Frankel set theory. That is, since ZF was invented. You don’t even need the Axiom of Choice. It’s provable with naive set theory too, or just with the Peano postulates and basic algebra.

Are you surprised? I was. What about non-Euclidean geometry? It turns out we can prove that too, and it’s all consistent (assuming ZF is consistent). How about two thousand years of mathematicians trying (and failing) to prove the parallel postulate? Did they just miss it? In one sense, yes, but in one sense no. Let’s dig a little deeper.
(more…)

Time for the annual wrap-up of what I saw where. Overall 2025 was a medium year. I spent about two months out of the country, though I didn’t take any major birding trips. I finished the year with 389 species total including 32 life birds including the very endangered and hard to find Ridgway’s Hawk:

• Green-breasted Mango – Juan Jose Flores Park, Heredia Costa Rica
• Stock Dove – St. Andrew Square Garden, Edinburgh Scotland
• Willow Warbler – Yorkshire
• Eurasian Skylark – The Coach House, Mesnes Lane, Leyburn Yorkshire
• Red-legged Partridge – The Coach House, Mesnes Lane, Leyburn Yorkshire
• Common Swift – Middleham | Yorkshire, England
• Reed Bunting – Marfield Wetlands NR | Yorkshire, England
• Rook – Marfield Wetlands NR | Yorkshire, England
• Red Kite Road from Skipton to Masham | Yorkshire, England
• Red Grouse – Road from Skipton to Masham | Yorkshire, England
• Little Gull – Jamaica Bay Wildlife Refuge–West Pond | New York
• Costa Rican Swift – Botanika Osa Peninsula | Costa Rica
• Slate-headed Tody-Flycatcher – La perica | Costa Rica
• Riverside Wren – Botanika Osa Peninsula | Costa Rica
• Gray-cowled Wood-Rail – Botanika Osa Peninsula | Costa Rica
• Ruddy-breasted Seedeater – finca san martin | Costa Rica
• Gray-headed Tanager – Ecoturístico La Tarde | Costa Rica
• Charming Hummingbird – Ecoturístico La Tarde | Costa Rica
• White-crested Coquette – Ecoturístico La Tarde | Costa Rica
• Great Curassow – Finca Miguel Sanchez, Rio Piro | Costa Rica
• Laughing Falcon – Vía 245, Golfito | Costa Rica
• Bare-throated Tiger-Heron – Vía 245, | Costa Rica
• Golden-naped Woodpecker – Botanika Osa Peninsula
• Greater Antillean Bullfinch – Westin Puntacana | Dominican Republic
• Hispaniolan Lizard-Cuckoo – Westin Puntacana | Dominican Republic
• Black-crowned Palm-Tanager – Punta Cana Ecological Reserve | Dominican Republic
• Stolid Flycatcher – Punta Cana Ecological Reserve | Dominican Republic
• Broad-billed Tody – Punta Cana Ecological Reserve | Dominican Republic
• Vervain Hummingbird – Punta Cana Ecological Reserve | Dominican Republic
• Hispaniolan Mango – Punta Cana Ecological Reserve | Dominican Republic
• Palmchat – Punta Cana | Dominican Republic
• Hispaniolan Woodpecker – Punta Cana | Dominican Republic
• Ridgway’s Hawk – Punta Cana | Dominican Republic

(more…)