If my botany knowledge is up to date, the 'seeds' you see there are actually the ovaries with the fruit on the inside. The fleshy part you eat is just receptacle tissue. I remember that from my botany course last year. If I find my notes I can confirm it for you. @silvermyth botanically speaking, it doesn't fit the definition of a botanical berry ("a fleshy fruit without a stone produced from a single flower containing one ovary"). A strawberry is an aggregate accessory fruit (it comes from more than one ovary and the tissue isn't all from the ovary). But it's still a fruit!
Also, the strawberries we find in our supermarkets are the product of selective breeding, a wild strawberry would fit the description of the berry. The fruit is the uterus of the plant, the seeds are like fertilized eggs in mammalian terms.
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